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17Calculus Integration - Weierstrass Substitution

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Weierstrass Substitution - Special Tangent Substitution

An interesting and special substitution that will often convert trig integrals into a form that can be integrated is to let \(t = \tan(x/2)\). From this we get the list of substitutions in the table below.

\(\displaystyle{ t = \tan(x/2) }\)

 

\(\displaystyle{ dx = \frac{2}{1+t^2}dt }\)

\(\displaystyle{ \sin(x) = \frac{2t}{1+t^2} }\)

 

\(\displaystyle{ \cos(x) = \frac{1-t^2}{1+t^2} }\)

Here is a short introduction of these equations.

blackpenredpen - weierstrass substitution for integrations, intro [7min-16secs]

video by blackpenredpen

This video shows the derivation of the equations. We recommend that you watch it, so that you will know where the equations come from and how to use them.

PatrickJMT - Integrate Rational Function of Sine and Cosine; t = tan(x/2), Part 1 [9min-42secs]

video by PatrickJMT

Okay, let's work some practice problems using this substitution.

How to Ace Calculus: The Streetwise Guide

Practice

Unless otherwise instructed, evaluate these integrals using Weierstrass substitution giving your answers in exact, simplified and factored form.

Basic

\(\displaystyle{\int{\frac{dx}{2\sin(x)+\sin(2x)}}}\)

Problem Statement

Evaluate the integral \(\displaystyle{\int{\frac{dx}{2\sin(x)+\sin(2x)}}}\). Give your answer in simplified, factored form.

Hint

This problem requires weierstrass substitution and trig integration to solve.

Problem Statement

Evaluate the integral \(\displaystyle{\int{\frac{dx}{2\sin(x)+\sin(2x)}}}\). Give your answer in simplified, factored form.

Final Answer

\(\displaystyle{\int{\frac{dx}{2\sin(x)+\sin(2x)}}}\) \(\displaystyle{ = \frac{1}{4}\ln[\tan(x/2)] + \frac{1}{8}\tan^2(x/2)+ C }\)

Problem Statement

Evaluate the integral \(\displaystyle{\int{\frac{dx}{2\sin(x)+\sin(2x)}}}\). Give your answer in simplified, factored form.

Hint

This problem requires weierstrass substitution and trig integration to solve.

Solution

To try to simplify this problem somewhat so that we can get some ideas, we use the identity \(\sin(2x) = 2\sin(x)\cos(x)\) in the denominator.

\(\displaystyle{ \int{ \frac{dx}{2\sin(x)+\sin(2x)} } }\)

\(\displaystyle{ \int{ \frac{dx}{2\sin(x)+2\sin(x)\cos(x)} } }\)

\(\displaystyle{ \int{ \frac{dx}{2\sin(x)(1+\cos(x))} } }\)

We could try substitution letting \(u=1+\cos(x)\) but that doesn't get us anywhere and no other basic substitution will either. So, let's try the substitution \(t=\tan(x/2)\). We know that
\(\displaystyle{ \sin(x) = \frac{2t}{1+t^2} }\) and \(\displaystyle{ \cos(x) = \frac{1-t^2}{1+t^2} }\)

From these, we can calculate expressions for \(1+\cos(x)\) and \(2\sin(x)(1+\cos(x))\).

\(\displaystyle{1+\cos(x) = 1 + \frac{1-t^2}{1+t^2} }\)

\(\displaystyle{ \frac{1+t^2}{1+t^2} + \frac{1-t^2}{1+t^2} }\)

\(\displaystyle{ \frac{1+t^2+1-t^2}{1+t^2} = \frac{2}{1+t^2} }\)

\(\displaystyle{ 2\sin(x)(1+\cos(x)) = 2 \left( \frac{2t}{1+t^2} \right) \left( \frac{2}{1+t^2} \right) = }\) \(\displaystyle{ \frac{8t}{(1+t^2)^2} }\)

Now we take the reciprocal of the last expression (since it is in the denominator of the integrand) substitute \(\displaystyle{ dx = \frac{2}{1+t^2}dt }\) and integrate.

\(\displaystyle{ \int{ \frac{(1+t^2)^2}{8t} \frac{2}{1+t^2} dt } }\)

\(\displaystyle{ \frac{1}{4}\int{ \frac{1+t^2}{t} dt } }\)

\(\displaystyle{ \frac{1}{4}\int{ \frac{1}{t} + t ~dt} }\)

\(\displaystyle{ \frac{1}{4} \left[ \ln(t) + \frac{t^2}{2} \right] + C }\)

\(\displaystyle{ \frac{1}{4} \ln[ \tan(x/2)] + \frac{1}{8}\tan^2 (x/2) + C }\)

Challenging Question: We checked our answer by using an online system and the answer they gave was
\(\displaystyle{\frac{1-2\cos^2(x/2)[ \ln(\cos(x/2)) - \ln(\sin(x/2)) ]}{4(\cos(x)+1)} + c_1}\)
Can you show that our answer is the same as this?

Final Answer

\(\displaystyle{\int{\frac{dx}{2\sin(x)+\sin(2x)}}}\) \(\displaystyle{ = \frac{1}{4}\ln[\tan(x/2)] + \frac{1}{8}\tan^2(x/2)+ C }\)

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\(\displaystyle{\int{\frac{dx}{3-5\sin(x)}}}\)

Problem Statement

Evaluate the integral \(\displaystyle{\int{\frac{dx}{3-5\sin(x)}}}\). Give your answer in simplified, factored form.

Hint

This problem requires weierstrass substitution and trig integration to solve.

Problem Statement

Evaluate the integral \(\displaystyle{\int{\frac{dx}{3-5\sin(x)}}}\). Give your answer in simplified, factored form.

Hint

This problem requires weierstrass substitution and trig integration to solve.

Solution

PatrickJMT - 124 video solution

video by PatrickJMT

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\(\displaystyle{ \int{ \frac{1}{1+\cos x + \sin x} ~dx } }\)

Problem Statement

Evaluate the integral \(\displaystyle{ \int{ \frac{1}{1+\cos x + \sin x} ~dx } }\). Give your answer in simplified, factored form.

Hint

This problem requires weierstrass substitution and trig integration to solve.

Problem Statement

Evaluate the integral \(\displaystyle{ \int{ \frac{1}{1+\cos x + \sin x} ~dx } }\). Give your answer in simplified, factored form.

Hint

This problem requires weierstrass substitution and trig integration to solve.

Solution

Integrals ForYou - 4281 video solution

Comment On Notation - Although his final answer is correct, he has some incorrect notation during the course of his solution. Notice that he doesn't include his constant of integration until the very end. To make the entire solution precisely correct, he needs to include the constant of integration in the step right after he does the actual integration. This is required since he writes equal signs between his steps. (This would also be required if he implied each step is equal to the previous one.) So don't do this or you may lose points for your work. However, as usual, check with your instructor to see what they require.

video by Integrals ForYou

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Intermediate

\(\displaystyle{ \int{ \frac{\cos x}{1+\cos x} ~dx } }\)

Problem Statement

Evaluate the integral \(\displaystyle{ \int{ \frac{\cos x}{1+\cos x} ~dx } }\). Give your answer in simplified, factored form.

Hint

This problem requires weierstrass substitution and trig integration to solve.

Problem Statement

Evaluate the integral \(\displaystyle{ \int{ \frac{\cos x}{1+\cos x} ~dx } }\). Give your answer in simplified, factored form.

Hint

This problem requires weierstrass substitution and trig integration to solve.

Solution

Integrals ForYou - 4282 video solution

Comment On Notation - Although his final answer is correct, he has some incorrect notation during the course of his solution. Notice that he doesn't include his constant of integration until the very end. To make the entire solution precisely correct, he needs to include the constant of integration in the step right after he does the actual integration. This is required since he writes equal signs between his steps. (This would also be required if he implied each step is equal to the previous one.) So don't do this or you may lose points for your work. However, as usual, check with your instructor to see what they require.

video by Integrals ForYou

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\(\displaystyle{ \int{ \frac{1}{1+\sin x}~dx } }\)

Problem Statement

Evaluate the integral \(\displaystyle{ \int{ \frac{1}{1+\sin x}~dx } }\). Give your answer in simplified, factored form.

Hint

This problem requires weierstrass substitution and trig integration to solve.

Problem Statement

Evaluate the integral \(\displaystyle{ \int{ \frac{1}{1+\sin x}~dx } }\). Give your answer in simplified, factored form.

Hint

This problem requires weierstrass substitution and trig integration to solve.

Solution

Integrals ForYou - 4283 video solution

Comment On Notation - Although his final answer is correct, he has some incorrect notation during the course of his solution. Notice that he doesn't include his constant of integration until the very end. To make the entire solution precisely correct, he needs to include the constant of integration in the step right after he does the actual integration. This is required since he writes equal signs between his steps. (This would also be required if he implied each step is equal to the previous one.) So don't do this or you may lose points for your work. However, as usual, check with your instructor to see what they require.

video by Integrals ForYou

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\(\displaystyle{ \int{ \frac{\sin^2 x}{(1+\cos x)^2} ~dx } }\)

Problem Statement

Evaluate the integral \(\displaystyle{ \int{ \frac{\sin^2 x}{(1+\cos x)^2} ~dx } }\). Give your answer in simplified, factored form.

Hint

This problem requires weierstrass substitution and trig integration to solve.

Problem Statement

Evaluate the integral \(\displaystyle{ \int{ \frac{\sin^2 x}{(1+\cos x)^2} ~dx } }\). Give your answer in simplified, factored form.

Hint

This problem requires weierstrass substitution and trig integration to solve.

Solution

Integrals ForYou - 4284 video solution

Comment On Notation - Although his final answer is correct, he has some incorrect notation during the course of his solution. Notice that he doesn't include his constant of integration until the very end. To make the entire solution precisely correct, he needs to include the constant of integration in the step right after he does the actual integration. This is required since he writes equal signs between his steps. (This would also be required if he implied each step is equal to the previous one.) So don't do this or you may lose points for your work. However, as usual, check with your instructor to see what they require.

video by Integrals ForYou

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Advanced

\(\displaystyle{ \int{ \frac{1}{\sin x} ~dx } }\)

Problem Statement

Evaluate the integral \(\displaystyle{ \int{ \frac{1}{\sin x} ~dx } }\). Give your answer in simplified, factored form.

Hint

This problem requires substitution and trig integration to solve. This problem looks like it should be easy but it isn't. There are at least four different ways to evaluate this integral, none of which may be obvious.
1. The weierstrass substitution \(t=\tan(x/2)\)
2. substitution \(u=x/2\)
3. substitution \(u=(1+\cos x)/(\sin x)\) after multiplying the integrand by \(u/u\)
4. start with \(\int{ \csc x ~dx}\) and let \(u=\csc x - \cot x\) after multiplying by \(u/u\)

Problem Statement

Evaluate the integral \(\displaystyle{ \int{ \frac{1}{\sin x} ~dx } }\). Give your answer in simplified, factored form.

Final Answer

\(\displaystyle{ \int{ \frac{1}{\sin x} ~dx } }\) \( = \ln\abs{ \tan(x/2) } + C_1 \) \( = \ln\abs{ \csc x - \cot x } + C_2 \)

Problem Statement

Evaluate the integral \(\displaystyle{ \int{ \frac{1}{\sin x} ~dx } }\). Give your answer in simplified, factored form.

Hint

This problem requires substitution and trig integration to solve. This problem looks like it should be easy but it isn't. There are at least four different ways to evaluate this integral, none of which may be obvious.
1. The weierstrass substitution \(t=\tan(x/2)\)
2. substitution \(u=x/2\)
3. substitution \(u=(1+\cos x)/(\sin x)\) after multiplying the integrand by \(u/u\)
4. start with \(\int{ \csc x ~dx}\) and let \(u=\csc x - \cot x\) after multiplying by \(u/u\)

Solution

This problem is solved several different ways as listed in the hint above.

Integrals ForYou - 4280 video solution

Comment On Notation - Although his final answer is correct, he has some incorrect notation during the course of his solution. Notice that he doesn't include his constant of integration until the very end. To make the entire solution precisely correct, he needs to include the constant of integration in the step right after he does the actual integration. This is required since he writes equal signs between his steps. (This would also be required if he implied each step is equal to the previous one.) So don't do this or you may lose points for your work. However, as usual, check with your instructor to see what they require.

video by Integrals ForYou

Integrals ForYou - 4280 video solution

Comment On Notation - Although his final answer is correct, he has some incorrect notation during the course of his solution. Notice that he doesn't include his constant of integration until the very end. To make the entire solution precisely correct, he needs to include the constant of integration in the step right after he does the actual integration. This is required since he writes equal signs between his steps. (This would also be required if he implied each step is equal to the previous one.) So don't do this or you may lose points for your work. However, as usual, check with your instructor to see what they require.

video by Integrals ForYou

Integrals ForYou - 4280 video solution

Comment On Notation - Although his final answer is correct, he has some incorrect notation during the course of his solution. Notice that he doesn't include his constant of integration until the very end. To make the entire solution precisely correct, he needs to include the constant of integration in the step right after he does the actual integration. This is required since he writes equal signs between his steps. (This would also be required if he implied each step is equal to the previous one.) So don't do this or you may lose points for your work. However, as usual, check with your instructor to see what they require.

video by Integrals ForYou

PatrickJMT - 4280 video solution

video by PatrickJMT

Final Answer

\(\displaystyle{ \int{ \frac{1}{\sin x} ~dx } }\) \( = \ln\abs{ \tan(x/2) } + C_1 \) \( = \ln\abs{ \csc x - \cot x } + C_2 \)

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\(\displaystyle{ \int{ \frac{\sin x}{1+\sin x} ~dx } }\)

Problem Statement

Evaluate the integral \(\displaystyle{ \int{ \frac{\sin x}{1+\sin x} ~dx } }\). Give your answer in simplified, factored form.

Hint

This problem requires weierstrass substitution and trig integration to solve.

Problem Statement

Evaluate the integral \(\displaystyle{ \int{ \frac{\sin x}{1+\sin x} ~dx } }\). Give your answer in simplified, factored form.

Hint

This problem requires weierstrass substitution and trig integration to solve.

Solution

Integrals ForYou - 4285 video solution

Comment On Notation - Although his final answer is correct, he has some incorrect notation during the course of his solution. Notice that he doesn't include his constant of integration until the very end. To make the entire solution precisely correct, he needs to include the constant of integration in the step right after he does the actual integration. This is required since he writes equal signs between his steps. (This would also be required if he implied each step is equal to the previous one.) So don't do this or you may lose points for your work. However, as usual, check with your instructor to see what they require.

video by Integrals ForYou

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\(\displaystyle{ \int{ \frac{1}{\cos x} ~dx } }\)

Problem Statement

Evaluate the integral \(\displaystyle{ \int{ \frac{1}{\cos x} ~dx } }\). Give your answer in simplified, factored form.

Hint

This problem requires substitution and trig integration to solve. This looks like it should be easy but it isn't. There are at least three different ways to evaluate the integral, none of which may be obvious.
1. The weierstrass substitution \(t=\tan(x/2)\)
2. start with \(\int{ \sec x ~dx}\) and let \(u=\sec x + \tan x \) after multiplying by \(u/u\)
3. substitution \(u=1/\cos x + \tan x\) after multiplying the integrand by \(u/u\)

Problem Statement

Evaluate the integral \(\displaystyle{ \int{ \frac{1}{\cos x} ~dx } }\). Give your answer in simplified, factored form.

Hint

This problem requires substitution and trig integration to solve. This looks like it should be easy but it isn't. There are at least three different ways to evaluate the integral, none of which may be obvious.
1. The weierstrass substitution \(t=\tan(x/2)\)
2. start with \(\int{ \sec x ~dx}\) and let \(u=\sec x + \tan x \) after multiplying by \(u/u\)
3. substitution \(u=1/\cos x + \tan x\) after multiplying the integrand by \(u/u\)

Solution

This problem is solved several different ways as listed in the hint above. The last two videos are by two different instructors that both use technique 3.

Integrals ForYou - 4286 video solution

Comment On Notation - Although his final answer is correct, he has some incorrect notation during the course of his solution. Notice that he doesn't include his constant of integration until the very end. To make the entire solution precisely correct, he needs to include the constant of integration in the step right after he does the actual integration. This is required since he writes equal signs between his steps. (This would also be required if he implied each step is equal to the previous one.) So don't do this or you may lose points for your work. However, as usual, check with your instructor to see what they require.

video by Integrals ForYou

Michel vanBiezen - 4286 video solution

video by Michel vanBiezen

Integrals ForYou - 4286 video solution

Comment On Notation - Although his final answer is correct, he has some incorrect notation during the course of his solution. Notice that he doesn't include his constant of integration until the very end. To make the entire solution precisely correct, he needs to include the constant of integration in the step right after he does the actual integration. This is required since he writes equal signs between his steps. (This would also be required if he implied each step is equal to the previous one.) So don't do this or you may lose points for your work. However, as usual, check with your instructor to see what they require.

video by Integrals ForYou

Michel vanBiezen - 4286 video solution

video by Michel vanBiezen

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\(\displaystyle{ \int{ \frac{1}{2 + \sin x} ~dx } }\)

Problem Statement

Evaluate the integral \(\displaystyle{ \int{ \frac{1}{2 + \sin x} ~dx } }\) using Weierstrass substitution giving your answers in exact, simplified and factored form.

Solution

Integrals ForYou - 4319 video solution

Comment On Notation - Although his final answer is correct, he has some incorrect notation during the course of his solution. Notice that he doesn't include his constant of integration until the very end. To make the entire solution precisely correct, he needs to include the constant of integration in the step right after he does the actual integration. This is required since he writes equal signs between his steps. (This would also be required if he implied each step is equal to the previous one.) So don't do this or you may lose points for your work. However, as usual, check with your instructor to see what they require.

video by Integrals ForYou

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Practice Instructions

Unless otherwise instructed, evaluate these integrals using Weierstrass substitution giving your answers in exact, simplified and factored form.

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