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17Calculus - Integration By Substitution

17Calculus
Single Variable Calculus
Derivatives
Integrals
Multi-Variable Calculus
Precalculus
Functions

Integration By Substitution

Integration by substitution is the first major integration technique that you will probably learn and it is the one you will use most of the time. In fact, as you learn more advanced techniques, you will still probably use this one also, in addition to the more advanced techniques, even on the same problem. So, this is a critically important technique to learn.

The main idea of this technique is to reduce the complexity of an integral by introducing a new variable to replace a part of the integrand. Then we convert the entire integrand in terms of the new variable. This should reduce the integral to one that we can evaluate using another technique.

Integration by substitution is sometimes called u-substitution since the letter \(u\) is often used as the substitution variable.

Topics You Need To Understand For This Page

[basics of integrals] - [differentials] - [substitution from precalculus]

If you want a complete lecture on integration by substitution, we recommend this video.

Prof Leonard - Calculus 1 Lecture 4.2: Integration by Substitution [1hr-33mins-57secs]

video by Prof Leonard

Where Integration by Substitution Comes From

The idea of integration of substitution comes from something you already now, the chain rule. Remember, the chain rule for \(y=f(g(x))\) looks like \(y'=f'(g(x))g'(x)\). Integration by substitution reverses this by first giving you \(f'(g(x))g'(x)\) and expecting you to come up with \(f(g(x))\). This is easier than you might think and it becomes easier as you get some experience. The key is to know how to choose \(g(x)\). First, let's see how the chain rule becomes an integral.

As we said above, the chain rule for \(y=f(g(x))\) is \(y'=f'(g(x))g'(x)\). This can also be written \(dy/dx=f'(g(x))g'(x)\) or in differential form \(dy=f'(g(x))g'(x)~dx\). Now when we integrate both sides we have \(y=\int{ f'(g(x))g'(x)dx }\). The key is to let \(u=g(x)\) and then \(du=g'(x)dx\). The integral can now be written \(y=\int{ f'(u)~du } = f(u)+C = f(g(x))+C\).

So the key to this technique is knowing what to choose for \(g(x)\). As mentioned above, the letter \(u\) is often used as the substitution variable, letting \(u=g(x)\).

How To Choose \( u \)

Let's watch a video clip with a VERY quick example. In this video she gives some really good advice on what to choose for \(u\).

Krista King Math - Solving Integrals [1min-39secs]

video by Krista King Math

Whew! That video clip went through the example pretty quickly without much explanation. Let's take it a bit slower and explain what is going on. In the video, the presenter evaluated the integral \(\int{x \sqrt{x^2+1} ~dx}\). Using her ideas, she chose \(u=x^2+1\). She actually lays out the solution very well but doesn't allow you much time to digest it. So, go back to the video and pause it at the moment she shows her complete solution and make sure you can follow each step.

Knowing what to choose for \(u\) is a skill that you get better at with practice. However, there are some specific guidelines to get you started. Let's go through them one-by-one and work some practice problems along the way to improve your skills.

Guideline 1 - Term Under A Power or Root

The video above gives one possible choice for \(u\), the higher power when you have another term with a power one less. From this example, I would extend that idea a bit to include choosing \(u\) as whatever is under a radical or a term with a power (like \((x^2+3)^3 \to u=x^2+3\)).

Let's look at an example to see how to apply this idea.

Evaluate \( \int{ 3x^2 \sqrt{x^3+1} ~dx} \).

Final Answer

\(\displaystyle{ \frac{2}{3}(x^3+1)^{3/2} +C }\)

Problem Statement

Evaluate \( \int{ 3x^2 \sqrt{x^3+1} ~dx} \).

Solution

Problem Statement: Evaluate \( \int{ 3x^2 \sqrt{x^3+1} ~dx} \).
Solution: Using the idea from the video above, we choose \(u=x^3+1 \to du=3x^2~dx \to dx=du/(3x^2)\).
Now we substitute \(u\) and dx into the integral.
\(\displaystyle{ \int{ 3x^2 \sqrt{x^3+1} ~dx} = \int{ 3x^2 \sqrt{u} \frac{du}{3x^2} } }\)
Now the \(3x^2\) term will cancel in the numerator and denominator leaving
\( \int{ u^{1/2}~du } \) which we evaluate to get
\(\displaystyle{ \frac{u^{3/2}}{3/2} + C = \frac{2u^{3/2}}{3} + C }\)
Now we need to go back to the original variable \(x\) using the original substitution \(u=x^3+1\)

Final Answer

\(\displaystyle{ \frac{2}{3}(x^3+1)^{3/2} +C }\)

Notice in the solution to the last example, that at one point we had \(x\)'s and \(u\)'s in the integral. We could not evaluate the integral until it had only the one variable \(u\). So had to cancel the \(3x^2\) term in the numerator and denominator before we could actually integrate.

Okay, now try some on your own. This is one technique that you want to practice a lot until you can do it in your sleep since you will be using it in almost every integral that you evaluate.

How to Ace Calculus: The Streetwise Guide

Practice

Unless otherwise instructed, evaluate these integrals using integration by substitution. Give your answers in simplified, factored form.

Basic

\(\displaystyle{ \int{(3x-5)^{17}dx} }\)

Problem Statement

Evaluate \(\displaystyle{ \int{(3x-5)^{17}dx} }\) using integration by substitution. Give your answer in simplified, factored form.

Solution

Krista King Math - 1010 video solution

video by Krista King Math

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\(\displaystyle{ \int{x\sqrt{x^2+9}~dx} }\)

Problem Statement

Evaluate \(\displaystyle{ \int{x\sqrt{x^2+9}~dx} }\) using integration by substitution. Give your answer in simplified, factored form.

Solution

Krista King Math - 1012 video solution

video by Krista King Math

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\(\displaystyle{ \int{x^2\sqrt{2x^3-4}~dx} }\)

Problem Statement

Evaluate \(\displaystyle{ \int{x^2\sqrt{2x^3-4}~dx} }\) using integration by substitution. Give your answer in simplified, factored form.

Solution

Krista King Math - 1013 video solution

video by Krista King Math

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\(\displaystyle{ \int{2x(x^2+1)^{30}~dx} }\)

Problem Statement

Evaluate \(\displaystyle{ \int{2x(x^2+1)^{30}~dx} }\) using integration by substitution. Give your answer in simplified, factored form.

Solution

PatrickJMT - 1017 video solution

video by PatrickJMT

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\(\displaystyle{ \int{2x(x^2+4)^{100}~dx} }\)

Problem Statement

Evaluate \(\displaystyle{ \int{2x(x^2+4)^{100}~dx} }\) using integration by substitution. Give your answer in simplified, factored form.

Solution

PatrickJMT - 1021 video solution

video by PatrickJMT

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\(\displaystyle{ \int{y\sqrt[5]{3y^2+1}~dy} }\)

Problem Statement

Evaluate \(\displaystyle{ \int{y\sqrt[5]{3y^2+1}~dy} }\) using integration by substitution. Give your answer in simplified, factored form.

Solution

PatrickJMT - 1318 video solution

video by PatrickJMT

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\(\displaystyle{ \int{(12x+8)(3x^2+4x+1)^9~dx} }\)

Problem Statement

Evaluate \(\displaystyle{ \int{(12x+8)(3x^2+4x+1)^9~dx} }\) using integration by substitution. Give your answer in simplified, factored form.

Solution

PatrickJMT - 1319 video solution

video by PatrickJMT

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Intermediate

\(\displaystyle{\int{ \frac{\ln x}{x} ~dx} }\)

Problem Statement

Evaluate \(\displaystyle{\int{ \frac{\ln x}{x} ~dx} }\) using integration by substitution. Give your answer in simplified, factored form.

Solution

The Organic Chemistry Tutor - 2583 video solution

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\(\displaystyle{ \int{\frac{1}{(4x+7)^6}dx} }\)

Problem Statement

Evaluate \(\displaystyle{ \int{\frac{1}{(4x+7)^6}dx} }\) using integration by substitution. Give your answer in simplified, factored form.

Solution

Krista King Math - 1011 video solution

video by Krista King Math

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\(\displaystyle{ \int{\frac{4}{(3x+1)^5}~dx} }\)

Problem Statement

Evaluate \(\displaystyle{ \int{\frac{4}{(3x+1)^5}~dx} }\) using integration by substitution. Give your answer in simplified, factored form.

Solution

PatrickJMT - 1317 video solution

video by PatrickJMT

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\(\displaystyle{ \int{\frac{4x}{\sqrt{x^2+1}}dx} }\)

Problem Statement

Evaluate \(\displaystyle{ \int{\frac{4x}{\sqrt{x^2+1}}dx} }\) using integration by substitution. Give your answer in simplified, factored form.

Solution

Krista King Math - 1016 video solution

video by Krista King Math

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\(\displaystyle{ \int{\frac{(\ln(x))^{10}}{x}dx} }\)

Problem Statement

Evaluate \(\displaystyle{ \int{\frac{(\ln(x))^{10}}{x}dx} }\) using integration by substitution. Give your answer in simplified, factored form.

Solution

Krista King Math - 1014 video solution

video by Krista King Math

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\(\displaystyle{\int{ \csc x \cot x \sqrt{1-\csc x}~dx} }\)

Problem Statement

Evaluate \(\displaystyle{\int{ \csc x \cot x \sqrt{1-\csc x}~dx} }\) using integration by substitution. Give your answer in simplified, factored form.

Final Answer

\((2/3)(1-\csc x)^{3/2}+C\)

Problem Statement

Evaluate \(\displaystyle{\int{ \csc x \cot x \sqrt{1-\csc x}~dx} }\) using integration by substitution. Give your answer in simplified, factored form.

Solution

PatrickJMT - 2106 video solution

video by PatrickJMT

Final Answer

\((2/3)(1-\csc x)^{3/2}+C\)

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\(\displaystyle{ \int{ \frac{x^3}{\sqrt[3]{x^4+1}} ~ dx } }\)

Problem Statement

\(\displaystyle{ \int{ \frac{x^3}{\sqrt[3]{x^4+1}} ~ dx } }\)

Solution

Michael Penn - 4136 video solution

video by Michael Penn

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Guideline 2 - Exponential

In the case of an exponential, choose \(u\) as the exponent. Since we know how to evaluate \(\int{e^u~du}\), this choice should significantly simplify the integral.

Practice

Unless otherwise instructed, evaluate these integrals using integration by substitution. Give your answers in simplified, factored form.

Basic

\(\displaystyle{ \int{e^{3x}dx} }\)

Problem Statement

Evaluate \(\displaystyle{ \int{e^{3x}dx} }\) using integration by substitution. Give your answer in simplified, factored form.

Final Answer

\(\displaystyle{ \int{e^{3x}dx} }\) \(\displaystyle{ = \frac{e^{3x}}{3} + C }\)

Problem Statement

Evaluate \(\displaystyle{ \int{e^{3x}dx} }\) using integration by substitution. Give your answer in simplified, factored form.

Solution

Integrals ForYou - 3806 video solution

video by Integrals ForYou

Final Answer

\(\displaystyle{ \int{e^{3x}dx} }\) \(\displaystyle{ = \frac{e^{3x}}{3} + C }\)

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\(\displaystyle{ \int{e^{5x}dx} }\)

Problem Statement

Evaluate \(\displaystyle{ \int{e^{5x}dx} }\) using integration by substitution. Give your answer in simplified, factored form.

Final Answer

\(\displaystyle{ \int{e^{5x}dx} }\) \(\displaystyle{ = \frac{e^{5x}}{5} + C }\)

Problem Statement

Evaluate \(\displaystyle{ \int{e^{5x}dx} }\) using integration by substitution. Give your answer in simplified, factored form.

Solution

Integrals ForYou - 4268 video solution

video by Integrals ForYou

Final Answer

\(\displaystyle{ \int{e^{5x}dx} }\) \(\displaystyle{ = \frac{e^{5x}}{5} + C }\)

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\(\displaystyle{ \int{e^{-x}dx} }\)

Problem Statement

Evaluate \(\displaystyle{ \int{e^{-x}dx} }\) using integration by substitution. Give your answer in simplified, factored form.

Final Answer

\(\displaystyle{\int{e^{-x}dx}=-e^{-x}+C}\)

Problem Statement

Evaluate \(\displaystyle{ \int{e^{-x}dx} }\) using integration by substitution. Give your answer in simplified, factored form.

Solution

Evaluate \(\displaystyle{ \int{e^{-x}dx} }\).
For this integral we need to use the technique of integration by substitution. Since we know that \(\displaystyle{\int{e^{t}dt}=e^t+C}\), we let \(u=-x\to du=-dx\to-du=dx\)
\( \begin{array}{rcl} \int{e^{-x}dx} & = & \int{e^u (-du) } \\ & = & \int{-e^u du} \\ & = & -e^u + C \\ & = & -e^{-x} + C \end{array}\)

Final Answer

\(\displaystyle{\int{e^{-x}dx}=-e^{-x}+C}\)

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\(\displaystyle{ \int{ x \cdot e^{x^2} ~ dx } }\)

Problem Statement

Evaluate \(\displaystyle{ \int{ x \cdot e^{x^2} ~ dx } }\) using integration by substitution. Give your answer in simplified, factored form.

Final Answer

\(\displaystyle{ \int{ x \cdot e^{x^2} ~ dx } }\) \(\displaystyle{ = \frac{1}{2}e^{x^2} + C }\)

Problem Statement

Evaluate \(\displaystyle{ \int{ x \cdot e^{x^2} ~ dx } }\) using integration by substitution. Give your answer in simplified, factored form.

Solution

Notation Comment - 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.

Integrals ForYou - 4271 video solution

video by Integrals ForYou

Final Answer

\(\displaystyle{ \int{ x \cdot e^{x^2} ~ dx } }\) \(\displaystyle{ = \frac{1}{2}e^{x^2} + C }\)

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\(\displaystyle{ \int{\frac{e^x+1}{e^x}dx} }\)

Problem Statement

Evaluate \(\displaystyle{ \int{\frac{e^x+1}{e^x}dx} }\) using integration by substitution. Give your answer in simplified, factored form.

Solution

PatrickJMT - 1018 video solution

video by PatrickJMT

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\(\displaystyle{ \int{\frac{e^{\sqrt{x}}}{\sqrt{x}}dx} }\)

Problem Statement

Evaluate \(\displaystyle{ \int{\frac{e^{\sqrt{x}}}{\sqrt{x}}dx} }\) using integration by substitution. Give your answer in simplified, factored form.

Solution

PatrickJMT - 1020 video solution

video by PatrickJMT

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\(\displaystyle{ \int{ \frac{e^{1/x^4}}{x^5} ~ dx } }\)

Problem Statement

Evaluate \(\displaystyle{ \int{ \frac{e^{1/x^4}}{x^5} ~ dx } }\) using integration by substitution. Give your answer in simplified, factored form.

Hint

This looks harder than it is. Just follow the technique outlined above for your choice of \(u\) and see where it takes you.

Problem Statement

Evaluate \(\displaystyle{ \int{ \frac{e^{1/x^4}}{x^5} ~ dx } }\) using integration by substitution. Give your answer in simplified, factored form.

Final Answer

\(\displaystyle{ \int{ \frac{e^{1/x^4}}{x^5} ~ dx } }\) \(\displaystyle{ = \frac{-1}{4}e^{1/x^4} + C }\)

Problem Statement

Evaluate \(\displaystyle{ \int{ \frac{e^{1/x^4}}{x^5} ~ dx } }\) using integration by substitution. Give your answer in simplified, factored form.

Hint

This looks harder than it is. Just follow the technique outlined above for your choice of \(u\) and see where it takes you.

Solution

Integrals ForYou - 1366 video solution

video by Integrals ForYou

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.

Final Answer

\(\displaystyle{ \int{ \frac{e^{1/x^4}}{x^5} ~ dx } }\) \(\displaystyle{ = \frac{-1}{4}e^{1/x^4} + C }\)

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Intermediate

\(\displaystyle{ \int{ (x^2+x)e^{2x^3+3x^2}~dx }}\)

Problem Statement

Evaluate \(\displaystyle{ \int{ (x^2+x)e^{2x^3+3x^2}~dx }}\) using integration by substitution. Give your answer in simplified, factored form.

Final Answer

\( (1/6)e^{2x^3+3x^2} + C \)

Problem Statement

Evaluate \(\displaystyle{ \int{ (x^2+x)e^{2x^3+3x^2}~dx }}\) using integration by substitution. Give your answer in simplified, factored form.

Solution

PatrickJMT - 2102 video solution

video by PatrickJMT

Final Answer

\( (1/6)e^{2x^3+3x^2} + C \)

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\(\displaystyle{ \int{e^{x+e^x}~dx} }\)

Problem Statement

Evaluate \(\displaystyle{ \int{e^{x+e^x}~dx} }\) using integration by substitution. Give your answer in simplified, factored form.

Solution

PatrickJMT - 1023 video solution

video by PatrickJMT

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Guideline 3 - Trig Functions

When sine and cosine functions are involved, choose \(u\) as one of them. When you take the differential, the other one will be part of the differential. For example, \(u=\sin(x) \to du=\cos(x)~dx\). Also, it sometimes helps to choose u as the angle of a trig function, if the angle is complicated. For example, if you have \(\sin(3t)\), let \(u=3t\).
The same rules apply for hyperbolic trig functions as well.

Practice

Unless otherwise instructed, evaluate these integrals using integration by substitution. Give your answers in simplified, factored form.

Basic

\(\displaystyle{ \int_1^4{ \frac{\cos(\sqrt{x})}{\sqrt{x}} ~dx } }\)

Problem Statement

Evaluate the integral \(\displaystyle{ \int_1^4{ \frac{\cos(\sqrt{x})}{\sqrt{x}} ~dx } }\)

Solution

JDS1854 - 4349 video solution

video by JDS1854

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\(\displaystyle{ \int{ \frac{\cos x}{\sin^5 x} dx } }\)

Problem Statement

Evaluate \(\displaystyle{ \int{ \frac{\cos x}{\sin^5 x} dx } }\) using integration by substitution. Give your answer in simplified, factored form.

Final Answer

\(\displaystyle{ \int{ \frac{\cos x}{\sin^5 x} dx } }\) \(\displaystyle{ = \frac{-1}{4\sin^4 x} + C }\)

Problem Statement

Evaluate \(\displaystyle{ \int{ \frac{\cos x}{\sin^5 x} dx } }\) using integration by substitution. Give your answer in simplified, factored form.

Solution

Integrals ForYou - 4269 video solution

video by Integrals ForYou

Final Answer

\(\displaystyle{ \int{ \frac{\cos x}{\sin^5 x} dx } }\) \(\displaystyle{ = \frac{-1}{4\sin^4 x} + C }\)

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\(\displaystyle{ \int{(\sin x) e^{\cos x}~dx} }\)

Problem Statement

Evaluate \(\displaystyle{ \int{(\sin x) e^{\cos x}~dx} }\) using integration by substitution. Give your answer in simplified, factored form.

Final Answer

\(-e^{\cos x} + C\)

Problem Statement

Evaluate \(\displaystyle{ \int{(\sin x) e^{\cos x}~dx} }\) using integration by substitution. Give your answer in simplified, factored form.

Solution

PatrickJMT - 2101 video solution

video by PatrickJMT

Final Answer

\(-e^{\cos x} + C\)

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\(\displaystyle{ \int{ \frac{\cos(3x)}{e^{\sin(3x)}} ~ dx } }\)

Problem Statement

\(\displaystyle{ \int{ \frac{\cos(3x)}{e^{\sin(3x)}} ~ dx } }\)

Solution

Michael Penn - 4135 video solution

video by Michael Penn

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

Problem Statement

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

Final Answer

\( 2\sqrt{1+\sin x} + C \)

Problem Statement

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

Solution

Here are a few intermediate steps that he skips in the video.

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

\(\displaystyle{ \int{ \frac{1}{x^{1/2}} ~ dx } }\)

\(\displaystyle{ \int x^{-1/2} ~ dx }\)

\(\displaystyle{ \frac{x^{1/2}}{1/2} }\)

\(\displaystyle{ 2x^{1/2} }\)

\(\displaystyle{ 2\sqrt{x} }\)

Integrals ForYou - 3783 video solution

video by Integrals ForYou

Final Answer

\( 2\sqrt{1+\sin x} + C \)

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\(\displaystyle{ \int{ \tan(ax+b) dx } }\)

Problem Statement

Evaluate \(\displaystyle{ \int{ \tan(ax+b) dx } }\) using integration by substitution. Give your answer in simplified, factored form.

Final Answer

\(\displaystyle{ \int{ \tan(ax+b) dx } }\) \(\displaystyle{ = (-1/a)\ln\abs{\cos(a+b)} + C }\)

Problem Statement

Evaluate \(\displaystyle{ \int{ \tan(ax+b) dx } }\) using integration by substitution. Give your answer in simplified, factored form.

Solution

Integrals ForYou - 4270 video solution

video by Integrals ForYou

Final Answer

\(\displaystyle{ \int{ \tan(ax+b) dx } }\) \(\displaystyle{ = (-1/a)\ln\abs{\cos(a+b)} + C }\)

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\(\displaystyle{ \int{\tanh(\omega t)~dt} }\)

Problem Statement

Evaluate \(\displaystyle{ \int{\tanh(\omega t)~dt} }\) using integration by substitution. Give your answer in simplified, factored form.

Final Answer

\(\displaystyle{\int{ \tanh(\omega t)~dt}=(1/\omega)\ln(\cosh(\omega t))+C }\)

Problem Statement

Evaluate \(\displaystyle{ \int{\tanh(\omega t)~dt} }\) using integration by substitution. Give your answer in simplified, factored form.

Solution

\(\displaystyle{\int{ \tanh(\omega t) ~dt } = \int{ \frac{\sinh(\omega t)}{\cosh(\omega t)} ~dt }}\)

\( u = \cosh(\omega t) \to du = \omega \sinh(\omega t) ~dt \)

\(\displaystyle{ \int{\frac{1}{u} \frac{du}{\omega}} }\)

\(\displaystyle{ \frac{1}{\omega} \ln|u| + C }\)

\(\displaystyle{ \frac{1}{\omega} \ln | \cosh(\omega t) | + C }\)

Since \( \cosh(\omega t) > 0 \) for all \(t\), we can drop the absolute value signs and write \(\ln | \cosh(\omega t) | \) as \(\ln ( \cosh(\omega t) ) \). However, leaving the absolute values signs is also correct.

Final Answer

\(\displaystyle{\int{ \tanh(\omega t)~dt}=(1/\omega)\ln(\cosh(\omega t))+C }\)

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

Problem Statement

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

Solution

Michael Penn - 4137 video solution

video by Michael Penn

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Intermediate

\(\displaystyle{ \int{ \left[ \sec(\cos x)\tan(\cos x )\right] \sin x ~dx } }\)

Problem Statement

Evaluate \(\displaystyle{ \int{ \left[ \sec(\cos x)\tan(\cos x )\right] \sin x ~dx } }\) using integration by substitution. Give your answer in simplified, factored form.

Final Answer

\(-\sec(\cos x) + C\)

Problem Statement

Evaluate \(\displaystyle{ \int{ \left[ \sec(\cos x)\tan(\cos x )\right] \sin x ~dx } }\) using integration by substitution. Give your answer in simplified, factored form.

Solution

PatrickJMT - 2105 video solution

video by PatrickJMT

Final Answer

\(-\sec(\cos x) + C\)

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Guideline 4 - Fraction Denominator

When it makes sense, sometimes choosing \(u\) as the denominator of a fraction is helpful since \(\int{(1/x)dx} = \ln(x)\). Let's look at an example.

Evaluate \(\displaystyle{ \int{ \frac{x^3}{5+x^4} ~dx} }\).

Final Answer

\(\displaystyle{ \frac{1}{4}\ln(5+x^4) + C }\)

Problem Statement

Evaluate \(\displaystyle{ \int{ \frac{x^3}{5+x^4} ~dx} }\).

Solution

Using one of the substitution suggestions above, set \(u\) to the denominator, i.e. \(u=5+x^4 \to du=4x^3dx\).
\(\displaystyle{ \int{ \frac{x^3}{5+x^4} ~dx} = \int{ \frac{x^3}{u}~\frac{du}{4x^3} } }\)
Now, the \(x^3\) terms cancel and we have an extra 4 in the denominator which we can pull outside the integral to get \(\displaystyle{ \frac{1}{4} \int{ \frac{1}{u}~du } = \frac{1}{4} \ln(u) + C }\)
Substituting back into \(x\) terms gives us the final answer.

Final Answer

\(\displaystyle{ \frac{1}{4}\ln(5+x^4) + C }\)

In the last example, notice that all the \(x\) terms canceled but we had an extra constant, which we factored out of the integral. This will happen quite often and it helps to move all the constants outside the integral like we did so that it is easy to see that all the variables in the integral are now \(u\)'s. Once we have that, we can do the actual integration.

Practice

Unless otherwise instructed, evaluate these integrals using integration by substitution. Give your answers in simplified, factored form.

Basic

\(\displaystyle{ \int{\frac{5x}{5+2x^2}dx} }\)

Problem Statement

Evaluate \(\displaystyle{ \int{\frac{5x}{5+2x^2}dx} }\) using integration by substitution. Give your answer in simplified, factored form.

Solution

Krista King Math - 1015 video solution

video by Krista King Math

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

Problem Statement

Evaluate \(\displaystyle{ \int{\frac{2^x}{2^x+1}dx} }\) using integration by substitution. Give your answer in simplified, factored form.

Solution

PatrickJMT - 1019 video solution

video by PatrickJMT

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\(\displaystyle{\int{ \frac{x}{x^2+4}~dx } }\)

Problem Statement

Evaluate \(\displaystyle{\int{ \frac{x}{x^2+4}~dx } }\) using integration by substitution. Give your answer in simplified, factored form.

Final Answer

\((1/2)\ln |x^2+4| + C\)

Problem Statement

Evaluate \(\displaystyle{\int{ \frac{x}{x^2+4}~dx } }\) using integration by substitution. Give your answer in simplified, factored form.

Solution

PatrickJMT - 2103 video solution

video by PatrickJMT

Final Answer

\((1/2)\ln |x^2+4| + C\)

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\(\displaystyle{ \int{ \frac{24x^3-4}{3x^4-2x+1}~dx } }\)

Problem Statement

Evaluate \(\displaystyle{ \int{ \frac{24x^3-4}{3x^4-2x+1}~dx } }\) using integration by substitution. Give your answer in simplified, factored form.

Final Answer

\(2\ln|3x^4-2x+1| + C\)

Problem Statement

Evaluate \(\displaystyle{ \int{ \frac{24x^3-4}{3x^4-2x+1}~dx } }\) using integration by substitution. Give your answer in simplified, factored form.

Solution

PatrickJMT - 2104 video solution

video by PatrickJMT

Final Answer

\(2\ln|3x^4-2x+1| + C\)

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Intermediate

\(\displaystyle{ \int{\frac{1}{x^2+4x+13}dx} }\)

Problem Statement

Evaluate \(\displaystyle{ \int{\frac{1}{x^2+4x+13}dx} }\) using integration by substitution. Give your answer in simplified, factored form.

Solution

Dr Chris Tisdell - 1846 video solution

video by Dr Chris Tisdell

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\(\displaystyle{ \int{ \frac{x}{1+x^4} ~dx } }\)

Problem Statement

\(\displaystyle{ \int{ \frac{x}{1+x^4} ~dx } }\)

Solution

blackpenredpen - 3561 video solution

video by blackpenredpen

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

Problem Statement

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

Solution

blackpenredpen - 3563 video solution

video by blackpenredpen

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Advanced

\(\displaystyle{ \int{ \frac{dx}{1+x^{1/4}} } }\)

Problem Statement

Evaluate \(\displaystyle{ \int{ \frac{dx}{1+x^{1/4}} } }\) using integration by substitution. Give your answer in simplified, factored form.

Solution

PatrickJMT - 1009 video solution

video by PatrickJMT

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\(\displaystyle{ \int{ \frac{x}{1-x^2+\sqrt{1-x^2}}dx } }\)

Problem Statement

Evaluate \(\displaystyle{ \int{ \frac{x}{1-x^2+\sqrt{1-x^2}}dx } }\) using integration by substitution. Give your answer in simplified, factored form.

Solution

PatrickJMT - 1024 video solution

video by PatrickJMT

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\(\displaystyle{ \int{ \frac{x}{x^2+6x+10}~dx } }\)

Problem Statement

Evaluate \(\displaystyle{ \int{ \frac{x}{x^2+6x+10}~dx } }\) using integration by substitution. Give your answer in simplified, factored form.

Solution

Dr Chris Tisdell - 1841 video solution

video by Dr Chris Tisdell

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\(\displaystyle{ \int{ \frac{1}{x^{1/2} + x^{1/4}} ~dx } }\)

Problem Statement

Evaluate the integral \(\displaystyle{ \int{ \frac{1}{x^{1/2} + x^{1/4}} ~dx } }\)

Hint

First, use substitution by letting \(u = x^{1/4}\). Once you have done the substitution, you will get an integral with a numerator of \(u^2\). Rewrite this as \( (u^2 - 1) + 1 \). Divide into two integrals and integrate separately.

Problem Statement

Evaluate the integral \(\displaystyle{ \int{ \frac{1}{x^{1/2} + x^{1/4}} ~dx } }\)

Hint

First, use substitution by letting \(u = x^{1/4}\). Once you have done the substitution, you will get an integral with a numerator of \(u^2\). Rewrite this as \( (u^2 - 1) + 1 \). Divide into two integrals and integrate separately.

Solution

Integrals ForYou - 4344 video solution

video by Integrals ForYou

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Where Integration by Substitution Comes From

How To Choose u

1 - Term Under A Power or Root - Practice

2 - Exponential - Practice

3 - Trig Functions - Practice

4 - Fraction Denominator - Practice

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Practice Instructions

Unless otherwise instructed, evaluate these integrals using integration by substitution. Give your answers in simplified, factored form.

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