Math Easy Solutions

In this video I go over the same example 5 on trig substitution for integrals which I did in my last video but this time solve it using hyperbolic trig substitution instead. Although using hyperbolic trig substitution in this example is more complicated, I wanted to illustrate that there were other methods of solving the same integral. The final answer we get using this method is in the form of an inverse hyperbolic trig function but when writing this function as a logarithmic function, we get the same answer as we did in my last video.

Download the notes in my video: http://1drv.ms/1PBxIFY

Related Videos:

Trigonometric Substitution for Integrals: Example 5: http://youtu.be/X-35GgXPEHQ
Trigonometric Substitution for Integrals: One to One Function: http://youtu.be/QO26MkCC3eU
Trigonometric Substitution for Integrals: Example 4: http://youtu.be/v3tHdU63Vio
Trigonometric Substitution for Integrals: Example 3: http://youtu.be/Tfl90TqW3jA
The Area of an Ellipse (and Circle): http://youtu.be/7hSkKPQA71s
Trigonometric Substitution for Integrals: Example 1: http://youtu.be/lpp6YLWB2GM
Trigonometric Substitution for Integrals: http://youtu.be/2pWvGXwtVJo
Inverse Functions Part 2: One to One Functions and the Horizontal Line Test: http://youtu.be/iSNuplUWrbA
Hyperbolic Functions - tanh(x), sinh(x), cosh(x) - Introduction: http://youtu.be/EmJKuQBEdlc
Inverse Hyperbolic Functions - Inverse sinh(x), cosh(x), tanh(x): http://youtu.be/lDmwSqRZOUA
Derivative of Inverse Hyperbolic Functions: inverse sinh(x), cosh(x), tanh(x): http://youtu.be/ubYbYbJOsNs .

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In this video I go over yet another example on trig substitution for integrals and this time solve the integral of the function x/sqrt(3-2x-x^2). In this example we first have to rewrite the function inside the square root by completing the square and using regular substitution to make it in the form we are familiar with to apply trig substitution.

Download the notes in my video: http://1drv.ms/1IPOOxY

View Video Notes on Steemit: https://steemit.com/mathematics/@mes/trigonometric-substitution-for-integrals-example-7

Related Videos:

Trigonometric Substitution for Integrals: Example 6: http://youtu.be/wh2dlrFbxrQ

Trigonometric Substitution for Integrals: Example 5: http://youtu.be/X-35GgXPEHQ
Trigonometric Substitution for Integrals: One to One Function: http://youtu.be/QO26MkCC3eU
Trigonometric Substitution for Integrals: Example 4: http://youtu.be/v3tHdU63Vio
Trigonometric Substitution for Integrals: Example 3: http://youtu.be/Tfl90TqW3jA
The Area of an Ellipse (and Circle): http://youtu.be/7hSkKPQA71s
Trigonometric Substitution for Integrals: Example 1: http://youtu.be/lpp6YLWB2GM
Trigonometric Substitution for Integrals: http://youtu.be/2pWvGXwtVJo
Completing the Square: http://youtu.be/5uxV6-f_qc0
Exact Trigonometry Ratios Part 1: 0, 30, 45, 60, and 90 Degrees: http://youtu.be/ln03a5KvQAY .

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In this video I over an introduction on partial fraction decomposition and go over some general techniques in decomposing a rational function that has unique linear factors in the denominator. Rational functions in this form are pretty straight forward to decompose but sometimes the denominator contains repeating and/or non-linear factors. For these cases the decomposition method is a bit more complicated and I will cover these in my later videos.

Download the notes in my video: https://1drv.ms/b/s!As32ynv0LoaIg9pp_a2r89J3CxLCIg

View Video Notes on Steemit: https://steemit.com/mathematics/@mes/partial-fraction-decomposition-general-techniques

Related Videos:

Types of Numbers: Natural, Integers, Rational, Irrational, and Real Numbers: http://youtu.be/U22Z1q_Ibqg
Polynomials - A Simple Explanation: http://youtu.be/IHIh7Y0kStE
Factoring Quadratic Polynomials by Guessing: http://youtu.be/biEfGwT5pn4
Factoring: Difference of Squares: http://youtu.be/qFdohskRQB0
Factoring: Difference and Addition of Cubes: http://youtu.be/LAUSBMod4Ks .

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In this video I go over further into partial fraction decomposition and show how to decompose rational functions that have non-linear factors (or irreducible factors) in the denominator. The method used for this case is similar to the technique used for linear factors but we have to account for the "size" of the numerator in the partial fractions so we need to apply write the numerator of each non-linear factor as a polynomial.

Download the notes in my video: http://1drv.ms/1z2g6OG

View Video Notes on Steemit: https://steemit.com/mathematics/@mes/partial-fraction-decomposition-non-linear-factors

Related Videos:

Partial Fraction Decomposition: Repeating Factors: http://youtu.be/9xlyx5NumkI
Partial Fraction Decomposition: General Techniques: http://youtu.be/bqf42x6nZoo
Types of Numbers: Natural, Integers, Rational, Irrational, and Real Numbers: http://youtu.be/U22Z1q_Ibqg
Polynomials - A Simple Explanation: http://youtu.be/IHIh7Y0kStE
Factoring Quadratic Polynomials by Guessing: http://youtu.be/biEfGwT5pn4
Factoring: Difference of Squares: http://youtu.be/qFdohskRQB0
Factoring: Difference and Addition of Cubes: http://youtu.be/LAUSBMod4Ks .

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In this video I go over a quick tutorial on how to decompose a rational function with both non-linear and repeating factors in the denominator. The rational function is pretty complicated so I just set up the partial fractions with the appropriate coefficients but did not solve for the coefficients as it would still be very tedious to solve them all. Basically in cases such as in this video, we can combine the techniques for decomposing non-linear functions as well as repeating factors which I covered in my earlier videos.

Download the notes in my video: http://1drv.ms/1ExaoEC

View Video Notes on Steemit: https://steemit.com/mathematics/@mes/partial-fraction-decomposition-non-linear-and-repeating-factors

Related Videos:

Partial Fraction Decomposition: Non-Linear Factors: http://youtu.be/ZK8akA4BE84
Partial Fraction Decomposition: Repeating Factors: http://youtu.be/9xlyx5NumkI
Partial Fraction Decomposition: General Techniques: http://youtu.be/bqf42x6nZoo
Types of Numbers: Natural, Integers, Rational, Irrational, and Real Numbers: http://youtu.be/U22Z1q_Ibqg
Polynomials - A Simple Explanation: http://youtu.be/IHIh7Y0kStE
Factoring Quadratic Polynomials by Guessing: http://youtu.be/biEfGwT5pn4
Factoring: Difference of Squares: http://youtu.be/qFdohskRQB0
Factoring: Difference and Addition of Cubes: http://youtu.be/LAUSBMod4Ks .

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In this video I cover how to decompose improper fractions. Improper fractions are fractions where the degree (or the highest power) of the denominator is smaller than or equal to the degree of the numerator. For improper fractions the techniques used in my earlier videos on proper fractions don't apply but we can first break down improper fractions to proper fractions by performing polynomial long division.

Download the notes in my video: http://1drv.ms/1GLWJMm

View Video Notes on Steemit: https://steemit.com/mathematics/@mes/partial-fraction-decomposition-improper-fractions

Related Videos:

Partial Fraction Decomposition: Non-Linear and Repeating Factors: http://youtu.be/wZVI5Gsc4WU
Partial Fraction Decomposition: Non-Linear Factors: http://youtu.be/ZK8akA4BE84
Partial Fraction Decomposition: Repeating Factors: http://youtu.be/9xlyx5NumkI
Partial Fraction Decomposition: General Techniques: http://youtu.be/bqf42x6nZoo
Types of Numbers: Natural, Integers, Rational, Irrational, and Real Numbers: http://youtu.be/U22Z1q_Ibqg
Polynomials - A Simple Explanation: http://youtu.be/IHIh7Y0kStE
Factoring Quadratic Polynomials by Guessing: http://youtu.be/biEfGwT5pn4
Factoring: Difference of Squares: http://youtu.be/qFdohskRQB0
Factoring: Difference and Addition of Cubes: http://youtu.be/LAUSBMod4Ks
Polynomial Long Division - In depth Look on why it works!: http://youtu.be/E1H584xJS_Y
Polynomial Long Division - Examples: http://youtu.be/7XbzCQgqBPc .

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In this video I go over polynomial long division and this time look at an example that involves multiple variables. The process of division is exactly the same as for single variables.

Download the notes in my video: http://1drv.ms/1IzYgEF

View Video Notes on Steemit: https://steemit.com/mathematics/@mes/polynomial-long-division-multiple-variables

Related Videos:

Polynomials - A Simple Explanation: http://youtu.be/IHIh7Y0kStE
Polynomial Long Division - In depth Look on why it works!: http://youtu.be/E1H584xJS_Y
Polynomial Long Division - Examples: http://youtu.be/7XbzCQgqBPc .

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In this video I go over further into partial fraction decomposition and this time explain why the techniques shown in my earlier videos for decomposing proper fractions don't work for improper fractions. Thus to decompose improper fractions we first need to perform polynomial long division to obtain proper fractions.

Download the notes in my video: http://1drv.ms/1F2bRTx

View Video Notes on Steemit: https://steemit.com/mathematics/@mes/partial-fraction-decomposition-why-it-doesn-t-for-improper-fractions

Related Videos:

Polynomial Long Division: Multiple Variables: http://youtu.be/vrElU5SR6Aw
Partial Fraction Decomposition: Non-Linear and Repeating Factors: http://youtu.be/wZVI5Gsc4WU
Partial Fraction Decomposition: Non-Linear Factors: http://youtu.be/ZK8akA4BE84
Partial Fraction Decomposition: Repeating Factors: http://youtu.be/9xlyx5NumkI
Partial Fraction Decomposition: General Techniques: http://youtu.be/bqf42x6nZoo
Types of Numbers: Natural, Integers, Rational, Irrational, and Real Numbers: http://youtu.be/U22Z1q_Ibqg
Polynomials - A Simple Explanation: http://youtu.be/IHIh7Y0kStE
Factoring Quadratic Polynomials by Guessing: http://youtu.be/biEfGwT5pn4
Factoring: Difference of Squares: http://youtu.be/qFdohskRQB0
Factoring: Difference and Addition of Cubes: http://youtu.be/LAUSBMod4Ks
Polynomial Long Division - In depth Look on why it works!: http://youtu.be/E1H584xJS_Y
Polynomial Long Division - Examples: http://youtu.be/7XbzCQgqBPc .

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In this video I go over a pretty extensive proof of the simplest case of partial fraction decomposition. The simplest case is for the case of a proper fraction without repeating or non-linear factors in the denominator of the rational function. The proof involves first proving a simpler theorem which allows us to use the proof of the simpler theorem to prove the overall partial fraction decomposition proof.

Download the notes in my video: http://1drv.ms/1KTIAeu

View Video Notes on Steemit: https://steemit.com/mathematics/@mes/partial-fraction-decomposition-simplest-case-proof

Related Videos:

Partial Fraction Decomposition: Why it doesn't for Improper Fractions: http://youtu.be/5QQP2kqE_ss
Partial Fraction Decomposition: Non-Linear and Repeating Factors: http://youtu.be/wZVI5Gsc4WU
Partial Fraction Decomposition: Non-Linear Factors: http://youtu.be/ZK8akA4BE84
Partial Fraction Decomposition: Repeating Factors: http://youtu.be/9xlyx5NumkI
Partial Fraction Decomposition: General Techniques: http://youtu.be/bqf42x6nZoo
Types of Numbers: Natural, Integers, Rational, Irrational, and Real Numbers: http://youtu.be/U22Z1q_Ibqg
Polynomials - A Simple Explanation: http://youtu.be/IHIh7Y0kStE
Factoring Quadratic Polynomials by Guessing: http://youtu.be/biEfGwT5pn4
Factoring: Difference of Squares: http://youtu.be/qFdohskRQB0
Factoring: Difference and Addition of Cubes: http://youtu.be/LAUSBMod4Ks .

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In this video, Mark aka MFA, goes over how to calculate the odds of winning a simple lottery. In the example used in the video the odds of winning the 649 lottery is shown. This lottery consists of selecting 6 numbers from a total of 49 possible numbers. The odds of guessing all 6 correct is about 1 in 14,000,000. A quick way of solving this is using the combination formula for probability and that is shown in the video too.

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View Video Notes on Steemit: https://steemit.com/mathematics/@mes/odds-of-winning-the-lottery

Related Videos:

Odds of Having a Perfect NCAA March Madness Bracket: http://youtu.be/It1sCq9cAFM
Probability that my friend Dmitry Scores Goals: http://youtu.be/zbAN04Kj3D8 .

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In this video I go over an extensive proof of decomposing rational functions for the general case with linear factors. Although the video is very long and detailed, it follows the same style as in my earlier proof on the simplest case in which a simple theorem was first proved in which it was repeatedly applied to the overall partial fraction decomposition proof. This is a very extensive and long video but it is worthwhile following along to get an idea of just how much detail is required in proving many of the math theorems out there.

Download the notes in my video: http://1drv.ms/1FsXyHW

View Video Notes on Steemit: https://steemit.com/mathematics/@mes/partial-fraction-decomposition-proof-of-general-case-with-linear-factors

Related Videos:

Partial Fraction Decomposition: Proof of the Simplest Case: http://youtu.be/Eq5yLJmC6pw
Partial Fraction Decomposition: Why it doesn't for Improper Fractions: http://youtu.be/5QQP2kqE_ss
Partial Fraction Decomposition: Non-Linear and Repeating Factors: http://youtu.be/wZVI5Gsc4WU
Partial Fraction Decomposition: Non-Linear Factors: http://youtu.be/ZK8akA4BE84
Partial Fraction Decomposition: Repeating Factors: http://youtu.be/9xlyx5NumkI
Partial Fraction Decomposition: General Techniques: http://youtu.be/bqf42x6nZoo
Types of Numbers: Natural, Integers, Rational, Irrational, and Real Numbers: http://youtu.be/U22Z1q_Ibqg
Polynomials - A Simple Explanation: http://youtu.be/IHIh7Y0kStE
Factoring Quadratic Polynomials by Guessing: http://youtu.be/biEfGwT5pn4
Factoring: Difference of Squares: http://youtu.be/qFdohskRQB0
Factoring: Difference and Addition of Cubes: http://youtu.be/LAUSBMod4Ks .

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In this video I summarize the basic linear and quadratic factors cases for partial fraction decomposition as well as talk briefly about factors with higher powers. It is also interesting to note that the 20 year old Evariste Galois proved that for factors with powers of 5 or higher there is no general formula for partial fraction decomposition nor will there ever exist any. Also of note is the Evariste died in a duel when he was only 20 years old.

Download the notes in my video: http://1drv.ms/1cohC2w

View Video Notes on Steemit: https://steemit.com/mathematics/@mes/partial-fraction-decomposition-summary-and-higher-powers

Related Videos:

Partial Fraction Decomposition: Proof of General Case with Linear Factors: https://youtu.be/crTTU9SpTNI
Partial Fraction Decomposition: Proof of the Simplest Case: http://youtu.be/Eq5yLJmC6pw
Partial Fraction Decomposition: Why it doesn't for Improper Fractions: http://youtu.be/5QQP2kqE_ss
Partial Fraction Decomposition: Non-Linear and Repeating Factors: http://youtu.be/wZVI5Gsc4WU
Partial Fraction Decomposition: Non-Linear Factors: http://youtu.be/ZK8akA4BE84
Partial Fraction Decomposition: Repeating Factors: http://youtu.be/9xlyx5NumkI
Partial Fraction Decomposition: General Techniques: http://youtu.be/bqf42x6nZoo
Types of Numbers: Natural, Integers, Rational, Irrational, and Real Numbers: http://youtu.be/U22Z1q_Ibqg
Polynomials - A Simple Explanation: http://youtu.be/IHIh7Y0kStE
Factoring Quadratic Polynomials by Guessing: http://youtu.be/biEfGwT5pn4
Factoring: Difference of Squares: http://youtu.be/qFdohskRQB0
Factoring: Difference and Addition of Cubes: http://youtu.be/LAUSBMod4Ks .

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In this video I go over an introduction on the method of partial fractions for integrating rational functions. Basically rational functions can be simplified by breaking them down to partial fractions using partial fraction decomposition which in turn greatly simplify the integration process. For an in depth look into partial fraction decomposition make sure to watch my earlier videos shown in the links below.

View Video Notes on Steemit: https://steemit.com/mathematics/@mes/integration-by-partial-fractions

Related Videos:

Partial Fraction Decomposition: Summary and Higher Powers: http://youtu.be/lSlkhrF09ww
Partial Fraction Decomposition: Proof of General Case with Linear Factors: https://youtu.be/crTTU9SpTNI
Partial Fraction Decomposition: Proof of the Simplest Case: http://youtu.be/Eq5yLJmC6pw
Partial Fraction Decomposition: Why it doesn't for Improper Fractions: http://youtu.be/5QQP2kqE_ss
Partial Fraction Decomposition: Non-Linear and Repeating Factors: http://youtu.be/wZVI5Gsc4WU
Partial Fraction Decomposition: Non-Linear Factors: http://youtu.be/ZK8akA4BE84
Partial Fraction Decomposition: Repeating Factors: http://youtu.be/9xlyx5NumkI
Partial Fraction Decomposition: General Techniques: http://youtu.be/bqf42x6nZoo
Area Under a Curve: Introduction to Integral Calculus: http://youtu.be/JbEbhv8ybmE .

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In this video I go over an example on using the method of partial fractions for integrating rational functions. In this example I go over the integral of (x^3+x)/(x-1) and break it down using polynomial long division. The remainder partial fraction is simple enough to integrate so we do not need to go any further in breaking it down using partial fraction decomposition.

Download the notes in my video: http://1drv.ms/1KBMIzo

View Video Notes on Steemit: https://steemit.com/mathematics/@mes/integration-by-partial-fractions-example-1

Related Videos:

Integration by Partial Fractions: http://youtu.be/r07NnKf76og
Partial Fraction Decomposition: Summary and Higher Powers: http://youtu.be/lSlkhrF09ww
Partial Fraction Decomposition: Proof of General Case with Linear Factors: https://youtu.be/crTTU9SpTNI
Partial Fraction Decomposition: Proof of the Simplest Case: http://youtu.be/Eq5yLJmC6pw
Partial Fraction Decomposition: Why it doesn't for Improper Fractions: http://youtu.be/5QQP2kqE_ss
Partial Fraction Decomposition: Non-Linear and Repeating Factors: http://youtu.be/wZVI5Gsc4WU
Partial Fraction Decomposition: Non-Linear Factors: http://youtu.be/ZK8akA4BE84
Partial Fraction Decomposition: Repeating Factors: http://youtu.be/9xlyx5NumkI
Partial Fraction Decomposition: General Techniques: http://youtu.be/bqf42x6nZoo
Area Under a Curve: Introduction to Integral Calculus: http://youtu.be/JbEbhv8ybmE
Polynomial Long Division - In depth Look on why it works!: http://youtu.be/E1H584xJS_Y
Polynomial Long Division - Examples: http://youtu.be/7XbzCQgqBPc .

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In this video I go over another example on integrating rational functions through the method of partial fractions and this time integrate the function (x^2+2x-1)/(2x^3+3x^2-2x). This requires using partial fraction decomposition and break down the rational function into simpler partial fractions which we know the integral of. Also in this video I allude to a theorem about inputting values even if the overall function is not defined for those variables. This theorem is actually a corollary to the continuity theorem and I will prove this in my next video so stay tuned for that!

Download the notes in my video: http://1drv.ms/1Jpb2HM

View Video Notes on Steemit: https://steemit.com/mathematics/@mes/integration-by-partial-fractions-example-2

Related Videos:

Integration by Partial Fractions: Example 1: http://youtu.be/UZthT_bCKyU
Integration by Partial Fractions: http://youtu.be/r07NnKf76og
Partial Fraction Decomposition: Summary and Higher Powers: http://youtu.be/lSlkhrF09ww
Partial Fraction Decomposition: Proof of General Case with Linear Factors: https://youtu.be/crTTU9SpTNI
Partial Fraction Decomposition: Proof of the Simplest Case: http://youtu.be/Eq5yLJmC6pw
Partial Fraction Decomposition: Why it doesn't for Improper Fractions: http://youtu.be/5QQP2kqE_ss
Partial Fraction Decomposition: Non-Linear and Repeating Factors: http://youtu.be/wZVI5Gsc4WU
Partial Fraction Decomposition: Non-Linear Factors: http://youtu.be/ZK8akA4BE84
Partial Fraction Decomposition: Repeating Factors: http://youtu.be/9xlyx5NumkI
Partial Fraction Decomposition: General Techniques: http://youtu.be/bqf42x6nZoo
Area Under a Curve: Introduction to Integral Calculus: http://youtu.be/JbEbhv8ybmE .

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In this video I go over another example on integrating rational functions by the method of partial fractions and this time solve the integral of 1/(x^2 - a^2) where a is not equal to 0. The result of this integral is a useful formula which can be used to simplify and solve integrals where a polynomial in the denominator can be written as a difference of squares.

Download the notes in my video: http://1drv.ms/1Mjr8jR

View Video Notes on Steemit: https://steemit.com/mathematics/@mes/integration-by-partial-fractions-example-3

Related Videos:

Corollary of Continuity Theorem: Proof: http://youtu.be/euAJAiFdIAA
Integration by Partial Fractions: Example 2: http://youtu.be/Fi16rcXUxHo
Integration by Partial Fractions: Example 1: http://youtu.be/UZthT_bCKyU
Integration by Partial Fractions: http://youtu.be/r07NnKf76og
Partial Fraction Decomposition: Summary and Higher Powers: http://youtu.be/lSlkhrF09ww
Partial Fraction Decomposition: Proof of General Case with Linear Factors: https://youtu.be/crTTU9SpTNI
Partial Fraction Decomposition: Proof of the Simplest Case: http://youtu.be/Eq5yLJmC6pw
Partial Fraction Decomposition: Why it doesn't for Improper Fractions: http://youtu.be/5QQP2kqE_ss
Partial Fraction Decomposition: Non-Linear and Repeating Factors: http://youtu.be/wZVI5Gsc4WU
Partial Fraction Decomposition: Non-Linear Factors: http://youtu.be/ZK8akA4BE84
Partial Fraction Decomposition: Repeating Factors: http://youtu.be/9xlyx5NumkI
Partial Fraction Decomposition: General Techniques: http://youtu.be/bqf42x6nZoo
Area Under a Curve: Introduction to Integral Calculus: http://youtu.be/JbEbhv8ybmE
Derivative of y = Log(x) and y = Ln(x): http://youtu.be/5e6MisvvMPE .

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In this video I go over another example on integrating rational functions by the method of partial fractions and this time go over an example that requires both polynomial long division and partial fraction decomposition to simplify and integrate. The example is solving the integral of (x^4-2x^2+4x+1)/(x^3-x^2-x+1).

Download the notes in my video: http://1drv.ms/1CNJjfY

View Video Notes on Steemit: https://steemit.com/mathematics/@mes/integration-by-partial-fractions-example-4

Related Videos:

Integration by Partial Fractions: Example 3: http://youtu.be/Ch2qP4uqluc
Corollary of Continuity Theorem: Proof: http://youtu.be/euAJAiFdIAA
Integration by Partial Fractions: Example 2: http://youtu.be/Fi16rcXUxHo
Integration by Partial Fractions: Example 1: http://youtu.be/UZthT_bCKyU
Integration by Partial Fractions: http://youtu.be/r07NnKf76og
Partial Fraction Decomposition: Summary and Higher Powers: http://youtu.be/lSlkhrF09ww
Partial Fraction Decomposition: Proof of General Case with Linear Factors: https://youtu.be/crTTU9SpTNI
Partial Fraction Decomposition: Proof of the Simplest Case: http://youtu.be/Eq5yLJmC6pw
Partial Fraction Decomposition: Why it doesn't for Improper Fractions: http://youtu.be/5QQP2kqE_ss
Partial Fraction Decomposition: Non-Linear and Repeating Factors: http://youtu.be/wZVI5Gsc4WU
Partial Fraction Decomposition: Non-Linear Factors: http://youtu.be/ZK8akA4BE84
Partial Fraction Decomposition: Repeating Factors: http://youtu.be/9xlyx5NumkI
Partial Fraction Decomposition: General Techniques: http://youtu.be/bqf42x6nZoo
Area Under a Curve: Introduction to Integral Calculus: http://youtu.be/JbEbhv8ybmE
Derivative of y = Log(x) and y = Ln(x): http://youtu.be/5e6MisvvMP Polynomial Long Division - Examples: http://youtu.be/7XbzCQgqBPc .

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In this video I go over another example on integration of rational functions by partial fractions and this time integrate a function that has a non-reducible non-linear factor in the denominator. This requires applying the method of partial fraction decomposition but for non-linear factors as I have shown in my earlier videos and you can see those in the links below. The example I go over is the integral of (2x^2 - x + 4)/(x^3 + 4x). In the example I derive the integral of 1/(x^2 + a^2) = 1/a * inverse tan (x/a) which was required in solving the overall integral.

Download the notes in my video: http://1drv.ms/1evrLwq

View Video Notes on Steemit: https://steemit.com/mathematics/@mes/integration-by-partial-fractions-example-5

Related Videos:

Integration by Partial Fractions: Example 4: http://youtu.be/79lWkqMEzUU

Integration by Partial Fractions: Example 3: http://youtu.be/Ch2qP4uqluc
Corollary of Continuity Theorem: Proof: http://youtu.be/euAJAiFdIAA
Integration by Partial Fractions: Example 2: http://youtu.be/Fi16rcXUxHo
Integration by Partial Fractions: Example 1: http://youtu.be/UZthT_bCKyU
Integration by Partial Fractions: http://youtu.be/r07NnKf76og
Partial Fraction Decomposition: Summary and Higher Powers: http://youtu.be/lSlkhrF09ww
Partial Fraction Decomposition: Proof of General Case with Linear Factors: https://youtu.be/crTTU9SpTNI
Partial Fraction Decomposition: Proof of the Simplest Case: http://youtu.be/Eq5yLJmC6pw
Partial Fraction Decomposition: Why it doesn't for Improper Fractions: http://youtu.be/5QQP2kqE_ss
Partial Fraction Decomposition: Non-Linear and Repeating Factors: http://youtu.be/wZVI5Gsc4WU
Partial Fraction Decomposition: Non-Linear Factors: http://youtu.be/ZK8akA4BE84
Partial Fraction Decomposition: Repeating Factors: http://youtu.be/9xlyx5NumkI
Partial Fraction Decomposition: General Techniques: http://youtu.be/bqf42x6nZoo
Area Under a Curve: Introduction to Integral Calculus: http://youtu.be/JbEbhv8ybmE
Derivative of y = Log(x) and y = Ln(x): http://youtu.be/5e6MisvvMP Polynomial Long Division - Examples: http://youtu.be/7XbzCQgqBPc
Derivative of arctan(x) or Inverse tan(x): http://youtu.be/q9jdOiQ7pZY .

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In this video I go over another example on integration of rational functions by partial fractions and this time go over an example that involves a non-linear function in the denominator that can't be factored any further. The example is on solving the integral of (4x^2-3x+2)/(4x^2-4x+3) and from this example we are able to obtain a general procedure for these types of integrals which end up being in the form of natural logs and inverse tangent functions.

Download the notes in my video: http://1drv.ms/1TSuUGx

View Video Notes on Steemit: https://steemit.com/mathematics/@mes/integration-by-partial-fractions-example-6

Related Videos:

Simple Proof of the Quadratic Formula: http://youtu.be/9eQLZar3g6g
Completing the Square: http://youtu.be/5uxV6-f_qc0
Integration by Partial Fractions: Example 5: http://youtu.be/ql2mu5joNJI
Integration by Partial Fractions: Example 4: http://youtu.be/79lWkqMEzUU
Integration by Partial Fractions: Example 3: http://youtu.be/Ch2qP4uqluc
Integration by Partial Fractions: Example 2: http://youtu.be/Fi16rcXUxHo
Integration by Partial Fractions: Example 1: http://youtu.be/UZthT_bCKyU
Integration by Partial Fractions: http://youtu.be/r07NnKf76og
Partial Fraction Decomposition: Summary and Higher Powers: http://youtu.be/lSlkhrF09ww
Partial Fraction Decomposition: Proof of General Case with Linear Factors: https://youtu.be/crTTU9SpTNI
Partial Fraction Decomposition: Proof of the Simplest Case: http://youtu.be/Eq5yLJmC6pw
Partial Fraction Decomposition: Why it doesn't for Improper Fractions: http://youtu.be/5QQP2kqE_ss
Partial Fraction Decomposition: Non-Linear and Repeating Factors: http://youtu.be/wZVI5Gsc4WU
Partial Fraction Decomposition: Non-Linear Factors: http://youtu.be/ZK8akA4BE84
Partial Fraction Decomposition: Repeating Factors: http://youtu.be/9xlyx5NumkI
Partial Fraction Decomposition: General Techniques: http://youtu.be/bqf42x6nZoo
Area Under a Curve: Introduction to Integral Calculus: http://youtu.be/JbEbhv8ybmE
Derivative of y = Log(x) and y = Ln(x): http://youtu.be/5e6MisvvMP Polynomial Long Division - Examples: http://youtu.be/7XbzCQgqBPc
Trigonometric Identity: Integral of 1/(x^2+a^2): https://youtu.be/XWsnhNCdzP8 .

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In this video I go over through some basics in economics and financing and discuss the Internal Rate of Return (IRR) as well as derive the formula for the Net Present Value (NPV). In financial budgeting and investment analysis, to compare different projects or investments the interest rate at which all current and future cash flows break even in terms of their present values is called the internal rate of return (IRR) and is simply an interest or discount rate. The Present Value (PV) is the value of any future cash flows but set to the date of valuation. The value of money increases with time so \$1 today is worth more than \$1 tomorrow due to the ability to gain value through interest. The NPV is simply the sum of all the PVs and when equal to 0, the corresponding interest rate is the IRR.

In this video I go over an example in which I derive the NPV formula and solve for the IRR using a Microsoft Excel spreadsheet to manipulate the IRR until the NPV = 0. This is a very useful introductory video to financing and investing so make sure to watch it!

View Video Notes on Steemit: https://steemit.com/economics/@mes/internal-rate-of-return-irr-and-net-present-value-npv

Related Videos:

Marginal Costs - Economics 101: http://youtu.be/XS-1L6Iq4Wk
Marginal Cost vs Average Cost - Economics 101: http://youtu.be/HiMaLvsTstc
Marginal Cost vs Average Cost Example - Economics 101: http://youtu.be/W2xx0Wtl608
Marginal Cost vs Marginal Revenue: When is maximum profit realized??: http://youtu.be/ufMgmYbZPdU
Marginal Cost vs Marginal Revenue - Examples Part 1: Maximizing Profit: http://youtu.be/cQ83F5qynco
Marginal Cost vs Marginal Revenue - Examples Part 2: Maximizing Revenue : http://youtu.be/nrH_Jrpm7vU .

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In this video I go over further into infinite sequences and series and this time look at the Power Series which is a series in which a power function is being summed up. The power series has many useful applications such as the vibrations of a drum as in the Bessel function and has to do with the particular behavior of the power series to oscillate in large amplitudes initially and dissipate with distance and/or time. I go over many examples, exercises, and proofs in yet another epic long math video, so follow along to learn more!

The topics and their timestamps in the video are listed below:

1. @ 1:54 - Power Series
- @ 9:41 - Note on Trigonometric Series and the Fourier Series
- @ 10:40 - Example 1
- @ 14:33 – Note on Factorials
- @ 15:23 - Example 2
2. @ 24:31 - Note on Bessel Functions
- @ 25:49 - Example 3: Bessel Function of Order 0
3. @ 49:41 - Theorem 1
- @ 55:54 - Summary of Previous Examples
- @ 58:35 - Example 4
- @ 1:07:22 - Example 5
4. Exercises
- @ 1:17:18 - Exercise 1: Bessel Function of Order 1
- @ 1:34:44 - Exercise 2: Proof of Theorem 1
- @ 1:35:53 - Theorem 2
- @ 1:49:24 - Theorem 3
- @ 1:53:48 - Completeness Axiom

View Video Notes on Steemit: https://steemit.com/mathematics/@mes/infinite-sequences-and-series-power-series

Related Videos:

Sequences and Series Playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0EXHAJ3vRg0T_kKEyPah1Lz

Infinite Sequences: Limits, Squeeze Theorem, Fibonacci Sequence & Golden Ratio + MORE: https://steemit.com/mathematics/@mes/infinite-sequences-limits-squeeze-theorem-fibonacci-sequence-and-golden-ratio-more

Infinite Series: Definition, Examples, Geometric Series, Harmonics Series, Telescoping Sum + MORE: https://steemit.com/mathematics/@mes/infinite-series-definition-examples-geometric-series-harmonics-series-telescoping-sum-more

Infinite Sequences and Series: The Integral Test and Estimate of Sums: https://steemit.com/mathematics/@mes/infinite-sequences-and-series-the-integral-test-and-estimate-of-sums

Infinite Sequences and Series: The Comparison Tests: https://steemit.com/mathematics/@mes/infinite-sequences-and-series-the-comparison-tests

Infinite Sequences and Series: Alternating Tests: https://steemit.com/mathematics/@mes/infinite-sequences-and-series-alternating-tests

Infinite Sequences and Series: Absolute Convergence and the Ratio/Root Tests: https://steemit.com/mathematics/@mes/infinite-sequences-and-series-absolute-convergence-and-the-ratio-root-tests

Infinite Sequences and Series: Strategy for Testing Series: https://steemit.com/mathematics/@mes/infinite-sequences-and-series-strategy-for-testing-series .

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In this video I go over another example on integrating rational functions by using partial fractions and this time solve the integral of the function (1-x+2x^2-x^3)/[x(x^2+1)^2]. The rational function in this example has a non-linear irreducible factor and thus I show how to break this down using partial fraction decomposition to simplify the integration process. This is a pretty tedious but useful video to fully understand the partial fractions process.

Download the notes in my video: http://1drv.ms/1SfZYfH

View Video Notes on Steemit: https://steemit.com/mathematics/@mes/integration-by-partial-fractions-example-7

Related Videos:

Integration by Partial Fractions: Example 6: http://youtu.be/3NqziB3knxE
Integration by Partial Fractions: Example 5: http://youtu.be/ql2mu5joNJI
Integration by Partial Fractions: Example 4: http://youtu.be/79lWkqMEzUU
Integration by Partial Fractions: Example 3: http://youtu.be/Ch2qP4uqluc
Integration by Partial Fractions: Example 2: http://youtu.be/Fi16rcXUxHo
Integration by Partial Fractions: Example 1: http://youtu.be/UZthT_bCKyU
Integration by Partial Fractions: http://youtu.be/r07NnKf76og
Partial Fraction Decomposition: Summary and Higher Powers: http://youtu.be/lSlkhrF09ww
Partial Fraction Decomposition: Proof of General Case with Linear Factors: https://youtu.be/crTTU9SpTNI
Partial Fraction Decomposition: Proof of the Simplest Case: http://youtu.be/Eq5yLJmC6pw
Partial Fraction Decomposition: Why it doesn't for Improper Fractions: http://youtu.be/5QQP2kqE_ss
Partial Fraction Decomposition: Non-Linear and Repeating Factors: http://youtu.be/wZVI5Gsc4WU
Partial Fraction Decomposition: Non-Linear Factors: http://youtu.be/ZK8akA4BE84
Partial Fraction Decomposition: Repeating Factors: http://youtu.be/9xlyx5NumkI
Partial Fraction Decomposition: General Techniques: http://youtu.be/bqf42x6nZoo
Area Under a Curve: Introduction to Integral Calculus: http://youtu.be/JbEbhv8ybmE
Derivative of y = Log(x) and y = Ln(x): http://youtu.be/5e6MisvvMP Polynomial Long Division - Examples: http://youtu.be/7XbzCQgqBPc
Trigonometric Identity: Integral of 1/(x^2+a^2): https://youtu.be/XWsnhNCdzP8 .

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In this video I go over yet another example on integrating rational functions by partial fractions but this time show how sometimes we can use integral substitution to avoid having to use partial fractions at all! The example I solve is the integral of the function (x^2+1)/[x*(x^2+3)] and use the substitution u = x*(x^2+3) to solve the integral without using partial fractions. This is a very useful technique as it can greatly simplify the integration process by avoiding partial fraction decomposition so it is always worth looking at different ways of solving any given integral!

Download the notes in my video: http://1drv.ms/1fAMOOD

View Video Notes on Steemit: https://steemit.com/mathematics/@mes/integration-by-partial-fractions-example-8-avoiding-partial-fractions

Related Videos:

Integration by Partial Fractions: Example 7: http://youtu.be/B_7YgHRNKvk

Integration by Partial Fractions: Example 6: http://youtu.be/3NqziB3knxE
Integration by Partial Fractions: Example 5: http://youtu.be/ql2mu5joNJI
Integration by Partial Fractions: Example 4: http://youtu.be/79lWkqMEzUU
Integration by Partial Fractions: Example 3: http://youtu.be/Ch2qP4uqluc
Integration by Partial Fractions: Example 2: http://youtu.be/Fi16rcXUxHo
Integration by Partial Fractions: Example 1: http://youtu.be/UZthT_bCKyU
Integration by Partial Fractions: http://youtu.be/r07NnKf76og
Partial Fraction Decomposition: Summary and Higher Powers: http://youtu.be/lSlkhrF09ww
Partial Fraction Decomposition: Proof of General Case with Linear Factors: https://youtu.be/crTTU9SpTNI
Partial Fraction Decomposition: Proof of the Simplest Case: http://youtu.be/Eq5yLJmC6pw
Partial Fraction Decomposition: Why it doesn't for Improper Fractions: http://youtu.be/5QQP2kqE_ss
Partial Fraction Decomposition: Non-Linear and Repeating Factors: http://youtu.be/wZVI5Gsc4WU
Partial Fraction Decomposition: Non-Linear Factors: http://youtu.be/ZK8akA4BE84
Partial Fraction Decomposition: Repeating Factors: http://youtu.be/9xlyx5NumkI
Partial Fraction Decomposition: General Techniques: http://youtu.be/bqf42x6nZoo
Area Under a Curve: Introduction to Integral Calculus: http://youtu.be/JbEbhv8ybmE
Derivative of y = Log(x) and y = Ln(x): http://youtu.be/5e6MisvvMP Polynomial Long Division - Examples: http://youtu.be/7XbzCQgqBPc .

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In this video I go over another example on integrating rational functions using partial fractions and this time show how sometimes even if the function we are integrating is a non-rational function we may be able to rationalize it or make it into a rational function using an appropriate substitution and thus be able to use partial fractions. The example I cover in this video is the integral of the function sqrt(x+4)/x and show how we can use the substitution u = sqrt(x+4) to transform the non-rational function into a rational function. This is a very useful technique to have when integrating non-rational functions so make sure to watch this video!

Download the notes in my video: http://1drv.ms/1O29Qsp

View Video Notes on Steemit: https://steemit.com/mathematics/@mes/integration-by-partial-fractions-example-9-rationalizing-substitutions

Related Videos:

Integration by Partial Fractions: Example 8: Avoiding Partial Fractions: http://youtu.be/32BibgNCNI4
Integration by Partial Fractions: Example 7: http://youtu.be/B_7YgHRNKvk
Integration by Partial Fractions: Example 6: http://youtu.be/3NqziB3knxE
Integration by Partial Fractions: Example 5: http://youtu.be/ql2mu5joNJI
Integration by Partial Fractions: Example 4: http://youtu.be/79lWkqMEzUU
Integration by Partial Fractions: Example 3: http://youtu.be/Ch2qP4uqluc
Integration by Partial Fractions: Example 2: http://youtu.be/Fi16rcXUxHo
Integration by Partial Fractions: Example 1: http://youtu.be/UZthT_bCKyU
Integration by Partial Fractions: http://youtu.be/r07NnKf76og
Partial Fraction Decomposition: Summary and Higher Powers: http://youtu.be/lSlkhrF09ww
Partial Fraction Decomposition: Proof of General Case with Linear Factors: https://youtu.be/crTTU9SpTNI
Partial Fraction Decomposition: Proof of the Simplest Case: http://youtu.be/Eq5yLJmC6pw
Partial Fraction Decomposition: Why it doesn't for Improper Fractions: http://youtu.be/5QQP2kqE_ss
Partial Fraction Decomposition: Non-Linear and Repeating Factors: http://youtu.be/wZVI5Gsc4WU
Partial Fraction Decomposition: Non-Linear Factors: http://youtu.be/ZK8akA4BE84
Partial Fraction Decomposition: Repeating Factors: http://youtu.be/9xlyx5NumkI
Partial Fraction Decomposition: General Techniques: http://youtu.be/bqf42x6nZoo
Area Under a Curve: Introduction to Integral Calculus: http://youtu.be/JbEbhv8ybmE
Derivative of y = Log(x) and y = Ln(x): http://youtu.be/5e6MisvvMP Polynomial Long Division - Examples: http://youtu.be/7XbzCQgqBP
Logarithms and their Properties - An Introduction: http://youtu.be/AZ6KKym19gI .

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In this video I go over an introduction to mortgages and the concept of amortization as well discuss a bit about our new online Mortgage Calculator where you can view at http://mc.mes.fm.

A mortgage is simply a loan given where the house or property is used as collateral which simply mean that you lose your house if you can’t pay the lender back. The concept of amortization is a method of paying back the loan gradually while compounding interest. This concept benefits both the lender and the homeowner since the lender will receive interest payments and the homeowner can pay a bit of the loan back at a time.

I also go over the proof of the formula for a basic amortization method where the interest rate is fixed and the monthly payments are constant. I further extend the proof to include a final balloon payment which is a way to save on interest by paying a big chunk of the mortgage at the end.

If you are looking to buy or sell your home it is very important to have a good understanding of mortgages and thus this is a very important video to watch!

Download the notes in my video: http://1drv.ms/1e3rEXK

View Video Notes on Steemit: https://steemit.com/mortgage/@mes/mortgage-amortization-formula-balloon-payment-proof

Related Videos:

Internal Rate of Return (IRR) and Net Present Value (NPV): http://youtu.be/2lGfXi_K3uE
Marginal Costs - Economics 101: http://youtu.be/XS-1L6Iq4Wk
Marginal Cost vs Average Cost - Economics 101: http://youtu.be/HiMaLvsTstc
Marginal Cost vs Average Cost Example - Economics 101: http://youtu.be/W2xx0Wtl608
Marginal Cost vs Marginal Revenue: When is maximum profit realized??: http://youtu.be/ufMgmYbZPdU
Marginal Cost vs Marginal Revenue - Examples Part 1: Maximizing Profit: http://youtu.be/cQ83F5qynco
Marginal Cost vs Marginal Revenue - Examples Part 2: Maximizing Revenue : http://youtu.be/nrH_Jrpm7vU
Final Exam Calculator (Updated) Tutorial: FinalExamCalculator.com: http://youtu.be/oNRb9am_nrk
GPA Calculator: finalexamcalculator.com/gpa: http://youtu.be/wEwOjGJmiDU
BMI (Body Mass Index) Introduction + History: http://youtu.be/4kuCuBjoXmE
Percentage Calculator: PercentCalculator.com (Update): https://youtu.be/wxbwPBve4sQ .

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