# 7.8: Homework- Integration by Parts

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$

$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$

( \newcommand{\kernel}{\mathrm{null}\,}\) $$\newcommand{\range}{\mathrm{range}\,}$$

$$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$

$$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$

$$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$

$$\newcommand{\Span}{\mathrm{span}}$$

$$\newcommand{\id}{\mathrm{id}}$$

$$\newcommand{\Span}{\mathrm{span}}$$

$$\newcommand{\kernel}{\mathrm{null}\,}$$

$$\newcommand{\range}{\mathrm{range}\,}$$

$$\newcommand{\RealPart}{\mathrm{Re}}$$

$$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$

$$\newcommand{\Argument}{\mathrm{Arg}}$$

$$\newcommand{\norm}[1]{\| #1 \|}$$

$$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$

$$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\AA}{\unicode[.8,0]{x212B}}$$

$$\newcommand{\vectorA}[1]{\vec{#1}} % arrow$$

$$\newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow$$

$$\newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vectorC}[1]{\textbf{#1}}$$

$$\newcommand{\vectorD}[1]{\overrightarrow{#1}}$$

$$\newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}}$$

$$\newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}}$$

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$

1. Solve each of the following using integration by parts:
1. $$\int x \cos(x)dx$$
$$x \sin(x) + \cos(x) + C$$
ans
2. $$\int (4x - 1) \cos(x)dx$$
$$(4 x - 1) \sin(x) + 4 \cos(x) + C$$
ans
3. $$\int x \sin(x)dx$$.
$$-x \cos(x) + \sin(x) + C$$
ans
4. $$\int x e^xdx$$.
$$x e^x - e^x + C$$
ans
5. $$\int \ln(x)dx$$. (Hint: Let $$u = \ln(x)$$ and $$v' = 1$$)
$$x \ln(x) - x$$
ans
2. Watch the following Khan Academy video: Integration by parts twice
3. Use integration by parts to solve $$\int x^2 \cos(x)dx$$.
$$x^2 \sin(x) + 2 x \cos(x) - 2\sin(x) + C$$
ans
4. Use integration by parts to solve $$\int x^3 e^xdx$$.
$$x^3 e^x - 3x^2 e^x + 6x e^x - 6 e^x + C$$
ans
5. Watch the following Khan Academy video: Integration by parts with e and cos together.
6. Use integration by parts to find $$\int e^x \sin(x)dx$$.
$$\frac{\sin(x) e^x - \cos(x) e^x}{2} + C$$
ans
7. Two part question:
1. Use $$u$$-substitution to find $$\int \sin(2x)dx$$ and $$\int \cos(2x)dx$$.
$$-\frac{1}{2} \cos(2x)$$ and $$\frac{1}{2} \sin(2x)$$
ans
2. Use integration by parts to find $$\int x \sin(2x)dx$$.
$$-\frac{1}{2} x \cos(2x) - \frac{1}{4} \sin(2x)$$
ans

This page titled 7.8: Homework- Integration by Parts is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by Tyler Seacrest via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.