3.1 Integration by parts  (Page 4/5)

${\displaystyle \int cos(\text{ln}x)dx}$

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

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

${\displaystyle \int x^2\text{ln}xdx}$

${\displaystyle \int {\text{sin}}^{\mathrm{-1}}xdx}$

${\displaystyle \int {\text{cos}}^{\mathrm{-1}}(2x)dx}$

${\displaystyle \int x\text{arctan}xdx}$

${\displaystyle \int x^2\text{sin}xdx}$

${\displaystyle \int x^3\text{cos}xdx}$

${\displaystyle \int x^3\text{sin}xdx}$

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

${\displaystyle \int x{\text{sec}}^{\mathrm{-1}}xdx}$

${\displaystyle \int x{\text{sec}}^{2}xdx}$

${\displaystyle \int x\text{cosh}xdx}$

${\displaystyle {\int }_{1\text{/}e}^1\text{ln}xdx}$

${\displaystyle {\int }_0^{1}x{e}^{\mathrm{-2}x}dx}$ (Express the answer in exact form.)

${\displaystyle {\int }_0^1{e}^{\sqrt{x}}dx\left(\text{let}u=\sqrt{x}\right)}$

${\displaystyle {\int }_1^e\text{ln}(x^2)dx}$

${\displaystyle {\int }_0^{\pi }x\text{cos}xdx}$

${\displaystyle {\int }_{\text{−}\pi }^{\pi }x\text{sin}xdx}$ (Express the answer in exact form.)

${\displaystyle {\int }_0^3\text{ln}(x^2+1)dx}$ (Express the answer in exact form.)

${\displaystyle {\int }_0^{\pi \text{/}2}{x}^2\text{sin}xdx}$ (Express the answer in exact form.)

${\displaystyle {\int }_0^{1}x{5}^{x}dx}$ (Express the answer using five significant digits.)

Evaluate ${\displaystyle \int \text{cos}x\text{ln}(\text{sin}x)dx}$

${\displaystyle \int x^n{e}^{x}dx=x^n{e}^x-n{\displaystyle \int x^{n-1}{e}^{x}dx}}$

${\displaystyle \int x^n\text{cos}xdx}=x^n\text{sin}x-n{\displaystyle \int x^{n-1}\text{sin}xdx}$

${\displaystyle \int x^n\text{sin}xdx}=\underset{}{\_\_\_\_\_\_}$

Integrate ${\displaystyle \int 2x\sqrt{2x-3}}dx$ using two methods:

1. Using parts, letting $dv=\sqrt{2x-3}dx$
2. Substitution, letting $u=2x-3$

${\displaystyle \int x\text{ln}xdx}$

${\displaystyle \int \frac{{\text{ln}}^{2}x}{x}dx}$

${\displaystyle \int x{e}^{x}dx}$

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

${\displaystyle \int x^2\text{sin}xdx}$

${\displaystyle \int x^2\text{sin}(3x^3+2)dx}$

$y=2x{e}^{\text{−}x}$ (Approximate answer to four decimal places.)

$y=e^{\text{−}x}\text{sin}(\pi x)$ (Approximate answer to five decimal places.)

$y=\text{sin}x,y=0,x=2\pi ,x=3\pi$ about the y-axis (Express the answer in exact form.)

$y=e^{\text{−}x}$ $y=0,x=\mathrm{-1}x=0;$ about $x=1$ (Express the answer in exact form.)

A particle moving along a straight line has a velocity of $v(t)=t^2{e}^{\text{−}t}$ after t sec. How far does it travel in the first 2 sec? (Assume the units are in feet and express the answer in exact form.)

Find the area under the graph of $y={\text{sec}}^{3}x$ from $x=0\text{to}x=1.$ (Round the answer to two significant digits.)

Find the area between $y=(x-2)e^x$ and the x- axis from $x=2$ to $x=5.$ (Express the answer in exact form.)

Find the area of the region enclosed by the curve $y=x\text{cos}x$ and the x -axis for

$\frac{11\pi }{2}\le x\le \frac{13\pi }{2}.$ (Express the answer in exact form.)

Find the volume of the solid generated by revolving the region bounded by the curve $y=\text{ln}x,$ the x -axis, and the vertical line $x=e^2$ about the x -axis. (Express the answer in exact form.)

Find the volume of the solid generated by revolving the region bounded by the curve $y=4\text{cos}x$ and the x -axis, $\frac{\pi }{2}\le x\le \frac{3\pi }{2},$ about the x -axis. (Express the answer in exact form.)

Find the volume of the solid generated by revolving the region in the first quadrant bounded by $y=e^x$ and the x -axis, from $x=0$ to $x=\text{ln}(7),$ about the y- axis. (Express the answer in exact form.)