Calculus 1 Practice Integrals
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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 Trig Integration (but not trig substitution) 

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
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
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 

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
video by blackpenredpen 

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 
\(\displaystyle{ \frac{1}{2} \csc^2 x + C }\) 
video by blackpenredpen 

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.
video by blackpenredpen 

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
video by Integrals ForYou 

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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
video by blackpenredpen 

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
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 \(u=4x \to du = 4dx\).
video by blackpenredpen 

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
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
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
video by blackpenredpen 

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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
video by Integrals ForYou 

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\(\displaystyle{ \int{ \frac{1}{(5x2)^4} ~ dx } }\)
Problem Statement
\(\displaystyle{ \int{ \frac{1}{(5x2)^4} ~ dx } }\)
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
video by Integrals ForYou 

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\(\displaystyle{ \int{ \frac{1}{x^34x^2} ~ dx } }\)
Problem Statement
\(\displaystyle{ \int{ \frac{1}{x^34x^2} ~ dx } }\)
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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\(\displaystyle{ \int_3^5{ (x3)^9 ~ dx } }\)
Problem Statement
\(\displaystyle{ \int_3^5{ (x3)^9 ~ dx } }\)
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 \( f(x) = \left\{ \begin{array}{ll} 10 & x \leq 2 \\ 3x^22 & x \gt 2 \end{array} \right. \)
Problem Statement
\(\displaystyle{ \int_0^5{ f(x) ~ dx } }\) for \( f(x) = \left\{ \begin{array}{ll} 10 & x \leq 2 \\ 3x^22 & x \gt 2 \end{array} \right. \)
Solution
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\(\displaystyle{ \int{ \frac{x1}{x^41} ~ dx } }\)
Problem Statement
\(\displaystyle{ \int{ \frac{x1}{x^41} ~ dx } }\)
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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\(\displaystyle{ \int{ \frac{\arcsin x}{\sqrt{1x^2}} ~ dx } }\)
Problem Statement
\(\displaystyle{ \int{ \frac{\arcsin x}{\sqrt{1x^2}} ~ dx } }\)
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
\(\displaystyle{ \int{ \frac{e^x \sqrt{e^x1}}{e^x+3} ~ dx } }\)
Problem Statement 

\(\displaystyle{ \int{ \frac{e^x \sqrt{e^x1}}{e^x+3} ~ dx } }\)
Hint 

We would usually use integration by substitution and let \(u=e^x1\) which is under the square root. However, the way he works it is to let \(u=\sqrt{e^x1}\). 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 

\(\displaystyle{ \int{ \frac{e^x \sqrt{e^x1}}{e^x+3} ~ dx } }\)
Final Answer 

\(\displaystyle{ 2\sqrt{e^x1} 4\arctan\left( \frac{\sqrt{e^x1}}{2} \right) + C }\)
Problem Statement
\(\displaystyle{ \int{ \frac{e^x \sqrt{e^x1}}{e^x+3} ~ dx } }\)
Hint
We would usually use integration by substitution and let \(u=e^x1\) which is under the square root. However, the way he works it is to let \(u=\sqrt{e^x1}\). 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
\(\displaystyle{ 2\sqrt{e^x1} 4\arctan\left( \frac{\sqrt{e^x1}}{2} \right) + C }\)
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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
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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 

\( \sqrt{1x^{2}} + C \)
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
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Final Answer
\( \sqrt{1x^{2}} + C \)
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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
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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
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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
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\(\displaystyle{ \int{ x^2 e^{x^3} ~ dx } }\)
Problem Statement
\(\displaystyle{ \int{ x^2 e^{x^3} ~ dx } }\)
Solution
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\(\int{ \tan x \ln( \cos x ) ~ dx }\)
Problem Statement
\(\int{ \tan x \ln( \cos x ) ~ dx }\)
Solution
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\(\displaystyle{ \int{ 2^{\ln x} ~ dx } }\)
Problem Statement
\(\displaystyle{ \int{ 2^{\ln x} ~ dx } }\)
Solution
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\(\displaystyle{ \int{ \frac{1}{1+\tan x} ~ dx } }\)
Problem Statement
\(\displaystyle{ \int{ \frac{1}{1+\tan x} ~ dx } }\)
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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\(\displaystyle{ \int{ \sqrt{ \frac{1x}{1+x} } ~ dx } }\)
Problem Statement 

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

Start by multiplying the numerator and denominator of the fraction by \(1x\).
Problem Statement
\(\displaystyle{ \int{ \sqrt{ \frac{1x}{1+x} } ~ dx } }\)
Hint
Start by multiplying the numerator and denominator of the fraction by \(1x\).
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
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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
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\(\int{ \sqrt{1+e^x} ~ dx }\)
Problem Statement
\(\int{ \sqrt{1+e^x} ~ dx }\)
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
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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
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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
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\(\displaystyle{ \int{ \frac{1}{\sqrt{xx^{3/2}}} ~ dx } }\)
Problem Statement
\(\displaystyle{ \int{ \frac{1}{\sqrt{xx^{3/2}}} ~ dx } }\)
Solution
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\(\int{ \sqrt{\tanh x} ~ dx }\)
Problem Statement
\(\int{ \sqrt{\tanh x} ~ dx }\)
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
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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
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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
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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