## 17Calculus Integrals - Calculus 1 Practice

##### 17Calculus

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.

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

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.

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

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

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

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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.

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

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

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

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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.

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

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

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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.

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

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

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

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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.

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

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

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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.

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

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

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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

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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.

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

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

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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.

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

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

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

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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.

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

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

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

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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

### blackpenredpen - 1747 video 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

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

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

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

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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

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

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

Problem Statement

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

Hint

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

Solution

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

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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

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

$$-\sqrt{1-x^{-2}} + C$$

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

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$$-\sqrt{1-x^{-2}} + C$$

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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

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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

### blackpenredpen - 1759 video 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

### blackpenredpen - 1760 video 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

### blackpenredpen - 3752 video 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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$$\int{ \csc x ~ dx }$$

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

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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

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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.

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

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.

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

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