$\int \frac{dx}{x^{2}+4x+20}$

$\int \frac{y+1}{y^{2}+y+1}\,dy$

$\int \sec ^{3}\phi \tan \phi \,d\phi$

$\int \cot^{2}\theta \,\csc \theta\, d\theta$

$\int \sin ^{3}\phi \,d\phi$

$\int \frac{x^{2}}{\sqrt{4-x^{2}}}dx$

$\int \frac{dx}{\sqrt{(x+2) ^{2}-1}}$

$\int_{0}^{\pi /4}x\sin (2x) \,dx$

$\int v\csc ^{2}v\,dv$

$\int\sin ^{2}x\cos ^{3}x\,dx$

$\int (4-x^{2})^{3/2}\,dx$

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

$\int \frac{e^{2t}\,dt}{e^{t}-2}$

$\int \frac{dy}{5+4y+4y^{2}}$

$\int \frac{x\,dx}{x^{4}-16}$

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

$\int \frac{y^{2}\,dy}{(y+1)^{3}}$

$\int \frac{dx}{x^{2}\sqrt{x^{2}+25}}$

$\int x\sec ^{2}x\,dx$

$\int \frac{dx}{\sqrt{16+4x-2x^{2}}}$

$\int \ln (1-y)\,dy$

$\int \frac{x^{3}-2x-1}{(x^{2}+1) ^{2}}dx$

$\int \frac{3x^{2}+2}{x^{3}-x^{2}}dx$

$\int \frac{dy}{\sqrt{2+3y^{2}}}$

$\int x^{2}\sin ^{-1}x\,dx$

$\int \sqrt{16+9x^{2}}\,dx$

$\int \frac{dx}{x^{2}+2x}$

$\int \sin ^{4}y\cos ^{4}y\,dy$

$\int \frac{w-2\,}{1-w^{2}}dw$

$\int \frac{x}{\sqrt{x^{2}-4}}dx$

$\int \frac{1}{\sqrt{x}}\cos ^{2}\sqrt{x}dx$

$\int \sin \left( \frac{\pi }{2}x\right) \sin (\pi x) \,dx$

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

$\int_{0}^{1}\frac{x^{2}}{\sqrt{4-x^{2}}}dx$

$\int_{0}^{\sqrt{3}}\frac{x\,dx}{\sqrt{1+x^{2}}}$

$\int x^{n}\,\tan ^{-1}x\,dx=\dfrac{x_{{}}^{n+1}}{n+1}\tan^{-1}x-\dfrac{1}{n+1}\int \dfrac{x^{n+1}}{1+x^{2}}dx$

$$\begin{align*} \int x^{n} (ax+b) ^{1/2}dx&=\dfrac{2x^{n}(ax+b) ^{3/2}}{(2n+3) a} \\ &\quad-\dfrac{2bn}{(2n+3) a} \int x^{n-1}(ax+b) ^{1/2}dx \end{align*}$$

$\int_{1}^{\infty }\dfrac{e^{-\sqrt{x}}}{\sqrt{x}}dx$

$\int_{0}^{1}\dfrac{\sin \sqrt{x}}{\sqrt{x}}dx$

$\int_{0}^{1}\dfrac{x\,dx}{\sqrt{1-x^{2}}}$

$\int_{-\infty }^{0}xe^{x}\,dx$

Show that $\int_{0}^{\pi /2}\dfrac{\sin x}{\cos x}\,dx$ diverges.

Show that $\int_{1}^{\infty }\dfrac{\sqrt{1+x^{1/8}}}{x^{3/4}}\,dx$ diverges.

$\int_{1}^{\infty }\dfrac{1+e^{-x}}{x}dx$

$\int_{0}^{\infty }\dfrac{x}{(1+x) ^{3}}dx$

If $\int x^{2}\cos x\,dx=f(x)-\int 2x\sin x\,dx$, find $f$.

Area and Volume

Arc Length Approximate the arc length of $y=\cos x$ from $x=0$ to $x=\dfrac{\pi }{2}$.

Distance The velocity $v$ (in meters per second) of a particle at time $t$ is given in the table. Use the Trapezoidal Rule to approximate the distance traveled from $t=1$ to $t=4.$

Area Find the area, if it exists, of the region bounded by the graphs of $y=x^{-2/3},$ $y=0,$ $x=0$, and $x=1.$

Volume Find the volume, if it exists, of the solid of revolution generated when the region bounded by the graphs of $y=x^{-2/3}$, $y=0,$ $x=0$, and $x=1$ is revolved about the $x$-axis.