\( \newcommand{\abs}[1]{\left| \, {#1} \, \right| } \) \( \newcommand{\cm}{\mathrm{cm} } \) \( \newcommand{\sec}{ \, \mathrm{sec} \, } \) \( \newcommand{\units}[1]{\,\text{#1}} \) \( \newcommand{\vhat}[1]{\,\hat{#1}} \) \( \newcommand{\vhati}{\,\hat{i}} \) \( \newcommand{\vhatj}{\,\hat{j}} \) \( \newcommand{\vhatk}{\,\hat{k}} \) \( \newcommand{\vect}[1]{\boldsymbol{\vec{#1}}} \) \( \newcommand{\norm}[1]{\|{#1}\|} \) \( \newcommand{\arccot}{ \, \mathrm{arccot} \, } \) \( \newcommand{\arcsec}{ \, \mathrm{arcsec} \, } \) \( \newcommand{\arccsc}{ \, \mathrm{arccsc} \, } \) \( \newcommand{\sech}{ \, \mathrm{sech} \, } \) \( \newcommand{\csch}{ \, \mathrm{csch} \, } \) \( \newcommand{\arcsinh}{ \, \mathrm{arcsinh} \, } \) \( \newcommand{\arccosh}{ \, \mathrm{arccosh} \, } \) \( \newcommand{\arctanh}{ \, \mathrm{arctanh} \, } \) \( \newcommand{\arccoth}{ \, \mathrm{arccoth} \, } \) \( \newcommand{\arcsech}{ \, \mathrm{arcsech} \, } \) \( \newcommand{\arccsch}{ \, \mathrm{arccsch} \, } \)

17Calculus Integrals - Calculus 1 Practice

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

Calculus 1 Practice Integrals

Integral Techniques Required To Solve These Problems

These practice problems are to prepare you for your calculus 1 integrals exam as well as your final exam in calculus 1. The techniques required for these problems are listed below. In your class, you may not have learned all of these techniques. If that is the case, ask your instructor for advice on which problems to work.

Basic Techniques

Definite Integrals

Both Fundamental Theorems [1st thm] - [2nd thm]

Integration by Substitution

Basic Trig Integration (but not trig substitution)

Inverse Trig Integration

Integration using Partial Fractions]

How to Read and Do Proofs: An Introduction to Mathematical Thought Processes

Practice

Unless otherwise instructed, evaluate these integrals giving your answers in exact, completely factored form.

Basic

\( \int{ x^{-3/2} ~ dx } \)

Problem Statement

Evaluate the integral \( \int{ x^{-3/2} ~ dx } \). Give your answer in simplified, factored form.

Hint

This problem requires only basic integration techniques to solved.

Problem Statement

Evaluate the integral \( \int{ x^{-3/2} ~ dx } \). Give your answer in simplified, factored form.

Final Answer

\( \int{ x^{-3/2} ~ dx } \) \(\displaystyle{ = -2x^{-1/2} + C }\)

Problem Statement

Evaluate the integral \( \int{ x^{-3/2} ~ dx } \). Give your answer in simplified, factored form.

Hint

This problem requires only basic integration techniques to solved.

Solution

Integrals ForYou - 4274 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

Final Answer

\( \int{ x^{-3/2} ~ dx } \) \(\displaystyle{ = -2x^{-1/2} + C }\)

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

Problem Statement

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

Hint

This problem can be solved with only basic integration techniques. However, a second solution is given that uses substitution.

Problem Statement

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

Final Answer

\(\displaystyle{ \int{ \frac{dx}{\sqrt{x}} } }\) \(\displaystyle{ = 2\sqrt{x} + C }\)

Problem Statement

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

Hint

This problem can be solved with only basic integration techniques. However, a second solution is given that uses substitution.

Solution

Integrals ForYou - 4273 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 - 4273 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

Final Answer

\(\displaystyle{ \int{ \frac{dx}{\sqrt{x}} } }\) \(\displaystyle{ = 2\sqrt{x} + C }\)

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

Problem Statement

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

Hint

This problem can be solved using basic trig integration.

Problem Statement

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

Final Answer

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

Problem Statement

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

Hint

This problem can be solved using basic trig integration.

Solution

blackpenredpen - 2615 video solution

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Final Answer

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

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\( \int{ \csc^3x \sec x ~ dx } \)

Problem Statement

Evaluate the integral \( \int{ \csc^3x \sec x ~ dx } \). Give your answer in simplified, factored form.

Hint

This problem requires basic trig integration and substitution to solve.

Problem Statement

Evaluate the integral \( \int{ \csc^3x \sec x ~ dx } \). Give your answer in simplified, factored form.

Final Answer

\( \int{ \csc^3x \sec x ~ dx } \) \(\displaystyle{ = \ln|\tan(x)| - \frac{1}{2}\csc^2 x + C }\)

Problem Statement

Evaluate the integral \( \int{ \csc^3x \sec x ~ dx } \). Give your answer in simplified, factored form.

Hint

This problem requires basic trig integration and substitution to solve.

Solution

A couple of comments are in order that will help clarify his solution.

Part way through this problem, he does a substitution, which is valid, but might not be obvious to you. He has the integral \[ \int{ \frac{1}{\sin^3 x \cos x} ~dx } \] For the next step, he uses the identity \(\sin^2 x + \cos^2 x = 1\) by replacing the one in the numerator with \(\sin^2 x + \cos^2 x\). Keep this substitution in mind as you progress through calculus. It can be handy in some cases.

For the third integral, he doesn't show the details using integration by substitution. Here they are.

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

Let \(u=\sin x \to du = \cos x ~ dx \)

\(\displaystyle{ \int{ \frac{du}{u^3} } }\)

\(\displaystyle{ \int{ u^{-3} ~ du} }\)

\(\displaystyle{ \frac{u^{-2}}{-2} + C }\)

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

\(\displaystyle{ \frac{-1}{2\sin^2 x} + C }\)

Since \(1/\sin x = \csc x\) this can be written

\(\displaystyle{ \frac{-1}{2} \csc^2 x + C }\)

blackpenredpen - 2617 video solution

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Final Answer

\( \int{ \csc^3x \sec x ~ dx } \) \(\displaystyle{ = \ln|\tan(x)| - \frac{1}{2}\csc^2 x + C }\)

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

Problem Statement

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

Hint

This problem requires integration by substitution to solve.

Problem Statement

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

Final Answer

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

Problem Statement

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

Hint

This problem requires integration by substitution to solve.

Solution

He doesn't show much work here to explain his solution. So here are the details.

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

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

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

Let \( u=-x/2 \to du = (-1/2)~dx \to -2du = dx \)\(\displaystyle{ }\)

\(\displaystyle{ \int{ e^u (-2du) } }\)

\(\displaystyle{ -2 \int{ e^u ~du } }\)

\(\displaystyle{ -2 e^u + C }\)

\(\displaystyle{ -2 e^{-x/2} + C }\)

Notice that right after he integrates, he leaves off the \(+C\). Although he adds it at the end of the problem and his answer is correct, it is incorrect to leave it off right after integrating. Make sure you don't make the same mistake. However, as usual, check with your instructor to see what they require.

blackpenredpen - 2619 video solution

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Final Answer

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

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

Problem Statement

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

Hint

This problem requires substitution to solve. This looks harder than it is. Try using the substitution \(u = 1/x^n\).

Problem Statement

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

Hint

This problem requires substitution to solve. This looks harder than it is. Try using the substitution \(u = 1/x^n\).

Solution

Integrals ForYou - 4277 video solution

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

Problem Statement

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

Hint

This problem requires basic trig integration and substitution to solve.

Problem Statement

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

Final Answer

\(\displaystyle{ \int{ \frac{2\sin x}{\sin(2x)} ~dx } }\) \(\displaystyle{ = \ln| \sec x + \tan x | + C }\)

Problem Statement

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

Hint

This problem requires basic trig integration and substitution to solve.

Solution

blackpenredpen - 2623 video solution

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Final Answer

\(\displaystyle{ \int{ \frac{2\sin x}{\sin(2x)} ~dx } }\) \(\displaystyle{ = \ln| \sec x + \tan x | + C }\)

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

Problem Statement

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

Hint

This problem requires substitution and trig integration to solve.

Problem Statement

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

Hint

This problem requires substitution and trig integration to solve.

Solution

Integrals ForYou - 4279 video solution

video by Integrals ForYou

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

Problem Statement

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

Hint

This problem can be solved using basic trig integration and substitution.

Problem Statement

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

Final Answer

\(\displaystyle{ \int{ \cos^2(2x) ~ dx } }\) \(\displaystyle{ = \frac{x}{2} + \frac{\sin(4x)}{8} + C }\)

Problem Statement

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

Hint

This problem can be solved using basic trig integration and substitution.

Solution

He does a simple substitution toward the end of the video clip where \(u=4x \to du = 4dx\).

blackpenredpen - 2624 video solution

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Final Answer

\(\displaystyle{ \int{ \cos^2(2x) ~ dx } }\) \(\displaystyle{ = \frac{x}{2} + \frac{\sin(4x)}{8} + C }\)

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

Problem Statement

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

Hint

This problem can be solved using basic integration techniques.

Problem Statement

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

Final Answer

\( \int{ (x+1/x)^2 ~ dx } \) \(\displaystyle{ = \frac{x^3}{3} + 2x -\frac{1}{x} + C }\)

Problem Statement

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

Hint

This problem can be solved using basic integration techniques.

Solution

blackpenredpen - 2627 video solution

video by blackpenredpen

Final Answer

\( \int{ (x+1/x)^2 ~ dx } \) \(\displaystyle{ = \frac{x^3}{3} + 2x -\frac{1}{x} + C }\)

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\(\displaystyle{ \int{ \frac{3}{x^2+4x+29} ~ dx } }\)

Problem Statement

Evaluate the integral \(\displaystyle{ \int{ \frac{3}{x^2+4x+29} ~ dx } }\). Give your answer in simplified, factored form.

Hint

This problem can be solved using trig integration and substitution techniques.

Problem Statement

Evaluate the integral \(\displaystyle{ \int{ \frac{3}{x^2+4x+29} ~ dx } }\). Give your answer in simplified, factored form.

Final Answer

\(\displaystyle{ \int{ \frac{3}{x^2+4x+29} ~ dx } }\) \(\displaystyle{ = \frac{3}{5} \arctan((x+2)/5) + C }\)

Problem Statement

Evaluate the integral \(\displaystyle{ \int{ \frac{3}{x^2+4x+29} ~ dx } }\). Give your answer in simplified, factored form.

Hint

This problem can be solved using trig integration and substitution techniques.

Solution

blackpenredpen - 2628 video solution

video by blackpenredpen

Final Answer

\(\displaystyle{ \int{ \frac{3}{x^2+4x+29} ~ dx } }\) \(\displaystyle{ = \frac{3}{5} \arctan((x+2)/5) + C }\)

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\( \int{ \sinh x ~ dx } \)

Problem Statement

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

Hint

This problem can be solved using basic integration techniques. Convert the integrand to exponentials and integrate directly.

Problem Statement

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

Hint

This problem can be solved using basic integration techniques. Convert the integrand to exponentials and integrate directly.

Solution

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\( \int{ \sinh^2 x ~ dx } \)

Problem Statement

Evaluate the integral \( \int{ \sinh^2 x ~ dx } \). Give your answer in simplified, factored form.

Hint

This problem requires substitution to solve.

Problem Statement

Evaluate the integral \( \int{ \sinh^2 x ~ dx } \). Give your answer in simplified, factored form.

Hint

This problem requires substitution to solve.

Solution

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\( \int{ \tanh x ~ dx } \)

Problem Statement

\( \int{ \tanh x ~ dx } \)

Solution

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\(\int{ \sech x ~ dx }\)

Problem Statement

\(\int{ \sech x ~ dx }\)

Solution

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\( \int{ (ax+b)^2 ~dx } \)

Problem Statement

Evaluate the integral \( \int{ (ax+b)^2 ~dx } \). Give your answer in simplified, factored form.

Hint

This problem requires substitution solve. You can avoid substitution by multiplying out before integrating but then you would have to factor a trinomial, which is not easy to see with the constants \(a\) and \(b\) in the polynomial. So we recommend using integration by substitution like he does in the video solution.

Problem Statement

Evaluate the integral \( \int{ (ax+b)^2 ~dx } \). Give your answer in simplified, factored form.

Hint

This problem requires substitution solve. You can avoid substitution by multiplying out before integrating but then you would have to factor a trinomial, which is not easy to see with the constants \(a\) and \(b\) in the polynomial. So we recommend using integration by substitution like he does in the video solution.

Solution

Integrals ForYou - 4278 video solution

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\(\displaystyle{ \int{ \frac{1}{(5x-2)^4} ~ dx } }\)

Problem Statement

\(\displaystyle{ \int{ \frac{1}{(5x-2)^4} ~ dx } }\)

Solution

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

Problem Statement

Evaluate the integral \( \int{ \tan(ax+b) ~ dx } \). Give your answer in simplified, factored form.

Hint

This problem requires substitution and trig integration to solve.

Problem Statement

Evaluate the integral \( \int{ \tan(ax+b) ~ dx } \). Give your answer in simplified, factored form.

Hint

This problem requires substitution and trig integration to solve.

Solution

Integrals ForYou - 4275 video solution

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

Problem Statement

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

Solution

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\(\int{ \sin x \cos(2x) ~ dx }\)

Problem Statement

\(\int{ \sin x \cos(2x) ~ dx }\)

Solution

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\(\int{ \sqrt{1+\cos(2x)} ~ dx }\)

Problem Statement

\(\int{ \sqrt{1+\cos(2x)} ~ dx }\)

Solution

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\(\displaystyle{ \int_3^5{ (x-3)^9 ~ dx } }\)

Problem Statement

\(\displaystyle{ \int_3^5{ (x-3)^9 ~ dx } }\)

Solution

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

Problem Statement

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

Solution

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

Problem Statement

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

Solution

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

Problem Statement

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

Solution

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\(\displaystyle{ \int_0^5{ f(x) ~ dx } }\) for \( f(x) = \left\{ \begin{array}{ll} 10 & x \leq 2 \\ 3x^2-2 & x \gt 2 \end{array} \right. \)

Problem Statement

\(\displaystyle{ \int_0^5{ f(x) ~ dx } }\) for \( f(x) = \left\{ \begin{array}{ll} 10 & x \leq 2 \\ 3x^2-2 & x \gt 2 \end{array} \right. \)

Solution

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

Problem Statement

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

Solution

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\(\displaystyle{ \int{ \frac{e^{\tan x}}{1-\sin^2 x} ~ dx } }\)

Problem Statement

\(\displaystyle{ \int{ \frac{e^{\tan x}}{1-\sin^2 x} ~ dx } }\)

Solution

In the video, he left out the work before his answer. Basically, he used integration by substitution with \( u = \tan x \).

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

Problem Statement

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

Solution

Although he doesn't write his work down, he does integration by substitution with \( u = \arctan x \)

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

Problem Statement

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

Solution

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

Problem Statement

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

Solution

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\(\displaystyle{ \int{ \frac{1}{\csc^3 x} ~ dx } }\)

Problem Statement

\(\displaystyle{ \int{ \frac{1}{\csc^3 x} ~ dx } }\)

Solution

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

Problem Statement

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

Solution

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

Problem Statement

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

Solution

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\(\int{ \sqrt[4]{x} ~ dx }\)

Problem Statement

\(\int{ \sqrt[4]{x} ~ dx }\)

Solution

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

Problem Statement

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

Solution

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Intermediate

\(\displaystyle{ \int{ \frac{e^x \sqrt{e^x-1}}{e^x+3} ~ dx } }\)

Problem Statement

\(\displaystyle{ \int{ \frac{e^x \sqrt{e^x-1}}{e^x+3} ~ dx } }\)

Hint

We would usually use integration by substitution and let \(u=e^x-1\) which is under the square root. However, the way he works it is to let \(u=\sqrt{e^x-1}\). It is a little more complicated at the first but the answer falls out nicely at the end when he integrates. So we suggest you do it his way.

Problem Statement

\(\displaystyle{ \int{ \frac{e^x \sqrt{e^x-1}}{e^x+3} ~ dx } }\)

Final Answer

\(\displaystyle{ 2\sqrt{e^x-1} -4\arctan\left( \frac{\sqrt{e^x-1}}{2} \right) + C }\)

Problem Statement

\(\displaystyle{ \int{ \frac{e^x \sqrt{e^x-1}}{e^x+3} ~ dx } }\)

Hint

We would usually use integration by substitution and let \(u=e^x-1\) which is under the square root. However, the way he works it is to let \(u=\sqrt{e^x-1}\). It is a little more complicated at the first but the answer falls out nicely at the end when he integrates. So we suggest you do it his way.

Solution

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Final Answer

\(\displaystyle{ 2\sqrt{e^x-1} -4\arctan\left( \frac{\sqrt{e^x-1}}{2} \right) + C }\)

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

Problem Statement

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

Hint

Factor \(\sqrt{x}\) out of the denominator term.

Problem Statement

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

Final Answer

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

Problem Statement

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

Hint

Factor \(\sqrt{x}\) out of the denominator term.

Solution

blackpenredpen - 2621 video solution

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Final Answer

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

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

Problem Statement

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

Hint

To get started, factor out an \(x^2\) from the square root and combine it with the \(x^2\) outside the square root. Then move that outside term to the numerator. Now you should be able to use integration by substitution.

Problem Statement

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

Final Answer

\( -\sqrt{1-x^{-2}} + C \)

Problem Statement

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

Hint

To get started, factor out an \(x^2\) from the square root and combine it with the \(x^2\) outside the square root. Then move that outside term to the numerator. Now you should be able to use integration by substitution.

Solution

blackpenredpen - 2631 video solution

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Final Answer

\( -\sqrt{1-x^{-2}} + C \)

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\(\displaystyle{ \int{ x^2 \sqrt[3]{1+x^3} ~ dx } }\)

Problem Statement

\(\displaystyle{ \int{ x^2 \sqrt[3]{1+x^3} ~ dx } }\)

Solution

blackpenredpen - 1302 video solution

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\(\displaystyle{ \int_0^5{ \left\lfloor x \right\rfloor ~ dx } }\)

Problem Statement

\(\displaystyle{ \int_0^5{ \left\lfloor x \right\rfloor ~ dx } }\)

Hint

\( \left\lfloor x \right\rfloor \) is the floor function.
This is kind of a trick question in that there is no closed form for the integral. You need to remember that the definite integral is the area under the curve. So draw the graph of this function and calculate the area directly.

Problem Statement

\(\displaystyle{ \int_0^5{ \left\lfloor x \right\rfloor ~ dx } }\)

Hint

\( \left\lfloor x \right\rfloor \) is the floor function.
This is kind of a trick question in that there is no closed form for the integral. You need to remember that the definite integral is the area under the curve. So draw the graph of this function and calculate the area directly.

Solution

blackpenredpen - 1755 video solution

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

Problem Statement

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

Solution

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

Problem Statement

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

Solution

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

Problem Statement

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

Solution

blackpenredpen - 3741 video solution

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

Problem Statement

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

Solution

blackpenredpen - 3744 video solution

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\(\int{ \tan x \ln( \cos x ) ~ dx }\)

Problem Statement

\(\int{ \tan x \ln( \cos x ) ~ dx }\)

Solution

blackpenredpen - 3746 video solution

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

Problem Statement

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

Solution

blackpenredpen - 3749 video solution

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

Problem Statement

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

Solution

blackpenredpen - 3751 video solution

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\(\displaystyle{ \int_{1/e}^{e}{ \frac{\sqrt{1-(\ln x)^2}}{x} ~ dx } }\)

Problem Statement

\(\displaystyle{ \int_{1/e}^{e}{ \frac{\sqrt{1-(\ln x)^2}}{x} ~ dx } }\)

Solution

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

Problem Statement

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

Hint

Start by multiplying the numerator and denominator of the fraction by \(1-x\).

Problem Statement

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

Hint

Start by multiplying the numerator and denominator of the fraction by \(1-x\).

Solution

blackpenredpen - 3758 video solution

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\(\displaystyle{ \int{ x^{x/\ln x} ~ dx } }\)

Problem Statement

\(\displaystyle{ \int{ x^{x/\ln x} ~ dx } }\)

Hint

Start by replacing the \(x\) in the base with \(e^{\ln x}\).

Problem Statement

\(\displaystyle{ \int{ x^{x/\ln x} ~ dx } }\)

Hint

Start by replacing the \(x\) in the base with \(e^{\ln x}\).

Solution

blackpenredpen - 3759 video solution

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

Problem Statement

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

Solution

blackpenredpen - 3765 video solution

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\(\int{ \sqrt{1+e^x} ~ dx }\)

Problem Statement

\(\int{ \sqrt{1+e^x} ~ dx }\)

Solution

blackpenredpen - 3780 video solution

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\(\displaystyle{ \int{ \frac{\sqrt{\tan x}}{\sin(2x)} ~ dx } }\)

Problem Statement

\(\displaystyle{ \int{ \frac{\sqrt{\tan x}}{\sin(2x)} ~ dx } }\)

Solution

blackpenredpen - 3781 video solution

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\(\displaystyle{ \int_0^{\pi/2}{ \frac{1}{\sin x} ~ dx } }\)

Problem Statement

\(\displaystyle{ \int_0^{\pi/2}{ \frac{1}{\sin x} ~ dx } }\)

Solution

blackpenredpen - 3782 video solution

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Advanced

\( \int{ \csc x ~ dx } \)

Problem Statement

\( \int{ \csc x ~ dx } \)

Hint

Multiply the numerator and denominator by \( \csc x - \cot x \), set this factor equal to \(u\) and try integration by substitution.

Problem Statement

\( \int{ \csc x ~ dx } \)

Hint

Multiply the numerator and denominator by \( \csc x - \cot x \), set this factor equal to \(u\) and try integration by substitution.

Solution

blackpenredpen - 1141 video solution

video by blackpenredpen

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\(\displaystyle{ \int{ \frac{1}{\sqrt{x-x^{3/2}}} ~ dx } }\)

Problem Statement

\(\displaystyle{ \int{ \frac{1}{\sqrt{x-x^{3/2}}} ~ dx } }\)

Solution

blackpenredpen - 1291 video solution

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\(\int{ \sqrt{\tanh x} ~ dx }\)

Problem Statement

\(\int{ \sqrt{\tanh x} ~ dx }\)

Solution

blackpenredpen - 1753 video solution

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

Problem Statement

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

Solution

blackpenredpen - 1763 video solution

video by blackpenredpen

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

Problem Statement

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

Solution

blackpenredpen - 3757 video solution

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

Problem Statement

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

Hint

1. Use the identity \(\sin(2x) = 2\sin x \cos x\)
2. Expand out \( (\sin x - \cos x)^2 \) and solve for \(2\sin x \cos x\). Use this in the denominator
3. Let \(u = \sin x - \cos x\) and perform integration by substitution

Problem Statement

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

Final Answer

\(\displaystyle{ \int{ \frac{\sin x + \cos x}{\sqrt{\sin(2x)}} ~dx } }\) \( = \arcsin( \sin x - \cos x ) + C\)

Problem Statement

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

Hint

1. Use the identity \(\sin(2x) = 2\sin x \cos x\)
2. Expand out \( (\sin x - \cos x)^2 \) and solve for \(2\sin x \cos x\). Use this in the denominator
3. Let \(u = \sin x - \cos x\) and perform integration by substitution

Solution

Integrals ForYou - 2334 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

Final Answer

\(\displaystyle{ \int{ \frac{\sin x + \cos x}{\sqrt{\sin(2x)}} ~dx } }\) \( = \arcsin( \sin x - \cos x ) + C\)

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