Antiderivatives by substitution of variables - AP Calculus AB

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Question

Solve the following integral using substitution:

$\int{xsin(x^2)dx}$

Answer

To solve the integral, we have to simplify it by using a variable to substitute for a variable of . For this problem, we will let u replace the expression . Next, we must take the derivative of . Its derivative is . Next, solve this equation for so that we may replace it in the integral. Plug in place of and $\frac{du}{2x}$ in place of into the original integral and simplify. The in the denominator cancels out the remaining in the integral, leaving behind a $\frac{1}{2}$ . We can pull the $\frac{1}{2}$ out front of the integral. Next, take the anti-derivative of the integrand and replace with the original expression, adding the constant to the answer. The specific steps are as follows:

1. $\int{xsin(x^2)dx}$

2.

3.

4. $\int{x(sinu)\frac{du}{2x}}$

5. $\frac{1}{2}\int{sinudu}$

6. $\frac{-1}{2}cosu+c$

7. $\frac{-1}{2}cos(x^2)+c$

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Question

Solve the following integral using substitution:

$\int{x^2e^{x^3}dx}$

Answer

To solve the integral, we have to simplify it by using a variable to substitute for a variable of . For this problem, we will let u replace the expression . Next, we must take the derivative of . Its derivative is . Next, solve this equation for so that we may replace it in the integral. Plug in place of and $\frac{du}{3x^2}$ in place of into the original integral and simplify. The in the denominator cancels out the remaining in the integral, leaving behind a $\frac{1}{3}$ . We can pull the $\frac{1}{3}$ out front of the integral. Next, take the anti-derivative of the integrand and replace with the original expression, adding the constant to the answer. The specific steps are as follows:

1. $\int{x^2e^{x^3}dx}$

2.

3.

4. $\int{x^2e^u(\frac{du}{3x^2})}$

5. $\frac{1}{3}\int{e^udu}$

6. $\frac{1}{3}e^u+c$

7. $\frac{1}{3}e^{x^3}+c$

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Question

Use a change of variable (aka a u-substitution) to evaluate the integral,

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

Answer

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

Integrals such as this are seen very commonly in introductory calculus courses. It is often useful to look for patterns such as the fact that the polynomial under the radical in our example, , happens to be one order higher than the factor outside the radical, You know that if you take a derivative of a second order polynomial you will get a first order polynomial, so let's define the variable:

(1)

Now differentiate with respect to to write the differential for ,

(2)

Looking at equation (2), we can solve for , to obtain $xdx =\frac{1}{2}du$ . Now if we look at the original integral we can rewrite in terms of

$\int x\sqrt{1+x^2}dx = \int\sqrt{1+x^2}\underbrace{xdx} = \int\sqrt{u}\left(\frac{1}{2}du \right )$

Now proceed with the integration with respect to .

$\int\sqrt{u}\left(\frac{1}{2}du \right ) = \frac{1}{2}\int u^{\frac{1}{2}}du$

$= \frac{1}{2}\left[\frac{1}{\frac{1}{2}+1}u^{\frac{1}{2}+1} \right ]+C$

$=\frac{1}{2} \left(\frac{2}{3}u^{\frac{3}{2}} \right )+C$

$=\frac{1}{2} \left(\frac{2}{3}u^{\frac{3}{2}} \right )+C$

$=\frac{1}{3}u^{\frac{3}{2}}+C$

Now write the result in terms of using equation (1), we conclude,

$\int x\sqrt{1+x^2}dx=\frac{1}{3}(x^2+1)^{\frac{3}{2}}+C$

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Question

Solve the following integral using substitution:

$\int{x^3(2+x^4)^5}dx$

Answer

To solve the integral, we have to simplify it by using a variable u to substitute for a variable of x.

For this problem, we will let u replace the expression .

Next, we must take the derivative of u. Its derivative is .

Next, solve this equation for dx so that we may replace it in the integral.

Plug in place of and $\frac{du}{4x^3}$ in place of into the original integral and simplify.

The in the denominator cancels out the remaining in the integral, leaving behind a $\frac{1}{4}$ . We can pull the $\frac{1}{4}$ out front of the integral. Next, take the anti-derivative of the integrand and replace u with the original expression, adding the constant to the answer.

The specific steps are as follows:

1. $\int{x^3(2+x^4)^5dx}$

2.

3.

4. $dx=\frac{du}{4x^3}$

5. $\int{x^3u^5(\frac{du}{4x^3})}$

6. $\frac{1}{4}\int{u^5}du$

7. $\frac{1}{4}(\frac{1}{6}u^6)$

8. $\frac{1}{4}(\frac{1}{6}(2+x^4)^6$

9. $\frac{1}{24}(2+x^4)^6+c$

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Question

Solve the following integral using substitution:

$\int{x^2\sqrt{x^3+1}}dx$

Answer

To solve the integral, we have to simplify it by using a variable to substitute for a variable of . For this problem, we will let u replace the expression . Next, we must take the derivative of u. Its derivative is . Next, solve this equation for so that we may replace it in the integral. Plug in place of and $\frac{du}{3x^2}$ in place of into the original integral and simplify. The in the denominator cancels out the remaining in the integral, leaving behind a $\frac{1}{3}$ . We can pull the $\frac{1}{3}$ out front of the integral. Next, take the anti-derivative of the integrand and replace with the original expression, adding the constant to the answer. The specific steps are as follows:

1. $\int{x^2\sqrt{x^3+1}}dx$

2. =

3.

4. $\int{x^2\sqrt{u}\frac{du}{3x^2}}$

5. $\frac{1}{3}\int{\sqrt{u}du}$

6. $\frac{1}{3}(\frac{2}{3}u^{\frac{3}{2}})+c$

7. $\frac{2}{9}u^{\frac{3}{2}}+c$

8. $\frac{2}{9}(x^3+1)^{\frac{3}{2}}+c$

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Question

Solve the following integral using substitution:

$\int{cos^3xsinxdx}$

Answer

To solve the integral, we have to simplify it by using a variable to substitute for a variable of . For this problem, we will let u replace the expression . Next, we must take the derivative of . Its derivative is . Next, solve this equation for so that we may replace it in the integral. Plug in place of and $\frac{du}{-sinx}$ in place of into the original integral and simplify. The in the denominator cancels out the remaining in the integral, leaving behind a . Next, take the anti-derivative of the integrand and replace with the original expression, adding the constant to the answer. The specific steps are as follows:

1. $\int{cos^3xsinxdx}$

2.

3.

4. $\int{u^3sinx(\frac{du}{-sinx})}$

5. $\int-u^3du$

6. $\frac{-1}{4}u^4+c$

7. $\frac{-cos^4x}{4}+c$

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Question

Solve the following integral using substitution:

$\int{e^xcos(e^x)dx}$

Answer

To solve the integral, we have to simplify it by using a variable to substitute for a variable of . For this problem, we will let u replace the expression . Next, we must take the derivative of . Its derivative is . Next, solve this equation for so that we may replace it in the integral. Plug in place of and $\frac{du}{e^x}$ in place of into the original integral and simplify. The in the denominator cancels out the remaining in the integral. Next, take the anti-derivative of the integrand and replace with the original expression, adding the constant to the answer. The specific steps are as follows:

1. $\int{e^xcos(e^x)dx}$

2.

3.

4. $\int{e^xcosu(\frac{du}{e^x})}$

5. $\int{cosudu}$

6.

7.

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Question

Solve the following integral using substitution:

$\int{\frac{sin(lnx)}{x}dx}$

Answer

To solve the integral, we have to simplify it by using a variable to substitute for a variable of . For this problem, we will let u replace the expression . Next, we must take the derivative of . Its derivative is $\frac{1}{x}$ . Next, solve this equation for so that we may replace it in the integral. Plug in place of and in place of into the original integral and simplify. The in the denominator cancels out the remaining in the integral. Next, take the anti-derivative of the integrand and replace with the original expression, adding the constant to the answer. The specific steps are as follows:

1. $\int{\frac{sin(lnx)}{x}dx}$

2.

3. $du=(\frac{1}{x}dx)$

4.

5. $\int{\frac{sinu}{x}dux}$

6. $\int{sinudu}$

7.

8.

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Question

Solve the following integral using substitution:

$\int{(x^2+e^{3x})(x^3+e^{3x})^{\frac{6}{7}}dx}$

Answer

To solve the integral, we have to simplify it by using a variable to substitute for a variable of . For this problem, we will let u replace the expression $x^3+e^{3x}$ . Next, we must take the derivative of . Its derivative is $3x^2+e^{3x}$ . Next, solve this equation for so that we may replace it in the integral. Plug in place of $x^3+e^{3x}$ and $\frac{du}{3x^2+e^{3x}}$ in place of into the original integral and simplify. The $x^2+e^{3x}$ in the denominator cancels out the remaining $x^2+e^{3x}$ in the integral, leaving behind a $\frac{1}{3}$ . We can pull the $\frac{1}{3}$ out front of the integral. Next, take the anti-derivative of the integrand and replace with the original expression, adding the constant to the answer. The specific steps are as follows:

1. $\int{(x^2+e^{3x})(x^3+e^{3x})^{\frac{6}{7}}dx}$

2. $u=x^3+e^{3x}$

3. $du=(3x^2+3e^{3x})dx$

4. $\int(x^2+e^{3x})u^{\frac{6}{7}(\frac{du}{x^2+e^{3x}})}$

5. $\frac{1}{3}\int{u^{\frac{6}{7}}}du$

6. $\frac{1}{3}(\frac{7}{13})u^{\frac{13}{7}}$

7. $\frac{7}{39}u^{\frac{13}{7}}+c$

8. $\frac{7}{39}(x^3+e^{3x})^{\frac{13}{7}}+c$

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Question

Use u-subtitution to fine

$\int \sin^2(x)\cos (x)dx$

Answer

Let

Then

Now we can subtitute

$\int sin^2(x) \cos(x) dx=\int u^2du= \frac{1}{3}u^3$

Now we substitute back

$\frac{1}{3}u^3=\frac{1}{3} \sin^3(x)$

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Question

Evaluate $\int_0^1 \frac{6x}{1+x^2} dx$

Answer

We can use substitution for this integral.

Let ,

then .

Multiplying this last equation by , we get .

Now we can make our substitutions

$\int_0^1 \frac{6x}{1+x^2} dx$ . Start

$=\int_1^2 \frac{3}{u}du$ . Swap out with , and with . Make sure you also plug the bounds on the integral into for to get the new bounds.

$=3\int_1^2 \frac{1}{u}du$ . Factor out the .

$=3 \ln(u)|_1^2$ . Integrate (absolute value signs are not needed since $1\le u \le2$ .)

$=3(\ln(2)-\ln(1))$ . Evaluate

$=3\ln2 -0 = 3\ln2$ .

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Question

$\int e^{4x-2}dx=?$

Answer

This is a u-substitution integral. We need to substitute the new function, which is modifying our base function (the exponential).

$\int e^xdx=e^x+C$ , but instead of that, our problem is . We can solve this integral by completing the substitution.

$u=4x-2.; du=4dx. ;dx=\frac{du}{4}.$

Now, we can replace everything in our integrand and rewrite in terms of our new variables:

$\int e^{4x-2}dx=\int e^u\frac{du}{4}=\frac{1}{4}\int e^udu.$

$=\frac{1}{4}e^u+C=\frac{1}{4}e^{4x-2}+C$ .

Remember to plug your variable back in and include the integration constant since we have an indefinite integral.

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Question

$\int 2cos(2x)dx=?$

Answer

This is a u-substitution integral. We need to choose the following substitutions:

Now, we can replace our original problem with our new variables:

$\int2cos(2x)dx=\int cos(2x)2dx=\int cos(u)du$

In the last step, we need to plug in our original function and add the integration constant.

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Question

$\int \sqrt{3+4x}dx=?$

Answer

This is a hidden u-substitution problem! Because we have a function under our square root, we cannot just simply integrate it. Therefore, we need to choose the function under the square root as our substitution variable!

$u=3+4x.; du=4dx.; dx=\frac{du}{4}.$

Now, let us rewrite our original equation in terms of our new variable!

$\int \sqrt{3+4x}dx=\int \sqrt{u}\frac{du}{4}=\frac{1}{4}\int \sqrt udu.$

$=\frac{1}{4}\int u^{1/2}=\frac{1}{4}\frac{u^{3/2}}{3/2}+C=\frac{1}{4}*\frac{2}{3}u^{3/2}+C$

$=\frac{2}{12}u^{3/2}+C=\frac{1}{6}({3+4x)}^{3/2}+C$ .

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Question

$\int \frac{2lnx}{x}dx=?$

Answer

This is a hidden u-substitution problem! Remember, to use substitution, we need to have an integral where a function and its derivative live inside. If you look closely, you will see we have just that!

$u=lnx.; du=\frac{1}{x}dx.$

Now, rewrite the integral, and integrate:

$\int \frac{2lnx}{x}==2\int lnx*\frac{1}{x}dx=2\int u du=2\frac{u^2}{2}+C$

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Question

$\int (2x-1)(x^2-x+8)^7dx=?$

Answer

This is a u-substitution problem. We need to find a function and its derivative in the integral.

Now, replace your variables, and integrate.

$\int (2x-1)(x^2-x+8)^7dx=\int (x^2-x+8)^7(2x-1)dx$

$=\int u^7du=\frac{u^8}{8}+C=\frac{(x^2-x+8)^8}{8}+C$ .

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Question

$\int x\sqrt{3+x}dx=?$

Answer

This problem is an application of the u-substitution method.

Now, be careful that you replace everything in the original integral in terms of our new variables. This includes the term!

$\int x\sqrt{3+x}dx=\int(u-3)u^{1/2}du=\int u^{3/2}-3u^{1/2}du$

$=\frac{u^{5/2}}{5/2}-3\frac{u^{3/2}}{3/2}=\frac{2}{5}u^{5/2}-2u^{3/2}+C$

$=\frac{2}{5}(3+x)^{5/2}-2(3+x)^{3/2}+C$ .

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Question

$\int \frac{2x-4}{2x^2-8x+3}dx=?$

Answer

To simplify the integral, we need to substitute new variables:

$\u=2x^2-8x+3.; \du=(4x-8)dx \du=2(2x-4)dx \\frac{1}{2}du=(2x-4)dx$

Now, we can replace our original variables, and integrate!

$\int \frac{2x-4}{2x^2-8x+3}dx=\int \frac{du}{2u}=\frac{1}{2}\int\frac{du}{u}$

$\=\frac{1}{2}ln|u|+C \\ =\frac{1}{2}ln{|2x^2-8x+3|}+C$ .

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Question

Integrate:

$\int 5y^2\cos(y^3)dy$

Answer

To integrate, we must make the following substitution:

The derivative was found using the following rule:

$\frac{\mathrm{d} }{\mathrm{d} x} x^n=nx^{n-1}$

Now, we rewrite the integral in terms of u and solve:

$\frac{5}{3}\int\cos(u)du=\frac{5}{3}\sin(u)+C$

The integral was found using the following rule:

$\int \cos(x)dx=\sin(x)+C$

Finally, replace u with our original x term:

$\frac{5}{3}\sin(y^3)+C$

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Question

Integrate:

$\int 3xe^{x^2}dx$

Answer

To integrate, we must perform the following substitution:

The derivative was found using the following rule:

$\frac{\mathrm{d} }{\mathrm{d} x} x^n=nx^{n-1}$

Now, we rewrite the integral in terms of u and solve:

$\frac{3}{2}\int\ e^udu=\frac{3}{2}e^u+C$

The integral was found using the following rule:

$\int e^xdx =e^x+C$

Finally, replace u with our original x term:

$\frac{3}{2}e^{x^2}+C$

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