Standard integrals

These formulas should be remembered for solving integral problems:

• $\int cf\left(x\right)dx=c\int f\left(x\right)dx$, where $c$ is a constant
• $\int \left\{f{(}_1\right)x\pm f_2\left(x\right)\}dx=\int f_1(x)dx\pm \int f_2(x)dx$
• $\int x^{n}dx=\frac{x^{n+1}}{n+1}$, where $n\ne -1$
• $\int \frac{1}{x}dx=\log x$
• $\int dx=x$
• $\int e^{x}dx=e^x$
• $\int e^{ax}dx=\frac{1}{a}{e}^{ax}$
• $\int a^{x}dx=\frac{a^x}{\log a}$, where $a>0$
• $\int \sin mxdx=-\frac{\cos mx}{m}$
• $\int \cos mxdx=\frac{\sin mx}{m}$
• $\int {\sec }^{2}xdx=\tan x$
• $\int {\csc }^{2}xdx=-\cot x$
• $\int \sec x\tan xdx=\sec x$
• $\int \csc x\cot xdx=-\csc x$
• $\int \sin \text{h}xdx=\cos \text{h}x$
• $\int \cos \text{h}xdx=\sin \text{h}x$
• $\int \sec {\text{h}}^{2}xdx=\tan \text{h}x$
• $\int \csc {\text{h}}^{2}xdx=-\cot \text{h}x$
• $\int \sec \text{h}x\tan \text{h}xdx=-\sec \text{h}x$
• $\int \csc \text{h}x\cot \text{h}xdx=-\csc \text{h}x$
• $\int \frac{f^{\prime }\left(x\right)}{f\left(x\right)}dx=\log \left(f\right(x\left)\right)$
• $\int \tan xdx=\log \left(\sec x\right)$
• $\int \cot xdx=\log \left(\sin x\right)$
• $\int \csc xdx=\log \left(\tan \frac{x}{2}\right)=\log (\csc x-\cot x)$
• $\int \sec xdx=\log \left\{\tan (\frac{\pi }{4}+\frac{x}{2})\right\}=\log (\sec x+\tan x)$
• $\int \frac{dx}{x^2+a^2}=\frac{1}{a}{\tan }^{-1}\frac{x}{a}$
• $\int \frac{dx}{x^2-a^2}=\frac{1}{2a}\log \frac{x-a}{x+a}$, where $x>a$
• $\int \frac{dx}{a^2-x^2}=\frac{1}{2a}\log \frac{a+x}{a-x}$, where $x<a$
• $\int \frac{dx}{\sqrt{a^2-x^2}}={\sin }^{-1}\frac{x}{a}$
• $\int \frac{dx}{\sqrt{x^2-a^2}}=\log (x+\sqrt{x^2-a^2})=\cos {\text{h}}^{-1}\frac{x}{a}$
• $\int \frac{dx}{\sqrt{x^2+a^2}}=\log (x+\sqrt{x^2+a^2})=\sin {\text{h}}^{-1}\frac{x}{a}$
• $\int \frac{dx}{x\sqrt{x^2-a^2}}=\frac{1}{a}{\sec }^{-1}\frac{x}{a}$
• $\int uvdx=u\int vdx-\int (\frac{du}{dx}.\int vdx)dx$
• $\int e^{ax}\cos bxdx=\frac{e^{ax}(a\cos bx+b\sin bx)}{a^2+b^2}=\frac{e^{ax}}{\sqrt{a^2+b^2}}\cos (bx-{\tan }^{-1}\frac{b}{a})$
• $\int e^{ax}\sin bxdx=\frac{e^{ax}(a\sin bx-b\cos bx)}{a^2+b^2}$
• $\int \sqrt{x^2+a^2}dx=\frac{x\sqrt{x^2+a^2}}{2}+\frac{a^2}{2}\log (x+\sqrt{x^2+a^2})=\frac{x\sqrt{x^2+a^2}}{2}+\frac{a^2}{2}\sin {\text{h}}^{-1}\frac{x}{a}$
• $\int \sqrt{x^2-a^2}dx=\frac{x\sqrt{x^2-a^2}}{2}-\frac{a^2}{2}\log (x+\sqrt{x^2-a^2})=\frac{x\sqrt{x^2-a^2}}{2}-\frac{a^2}{2}\cos {\text{h}}^{-1}\frac{x}{a}$
• $\int \sqrt{a^2-x^2}dx=\frac{x\sqrt{a^2-x^2}}{2}+\frac{a^2}{2}{\sin }^{-1}\frac{x}{a}$
• $\int e^x\left\{f\right(x)+f^{\prime }(x\left)dx\right\}=e^{x}f\left(x\right)$

5 Methods of integration

• Integration by substitution
• Integration by parts
• Decomposition into a sum
• Integration by successive reduction

6 Properties of definite integrals

$$\begin{array}{c}\begin{array}{cc}1.& {\int }_a^{b}f\left(x\right)dx={\int }_a^{b}f\left(t\right)dt\\ 2.& {\int }_a^{b}f\left(x\right)dx=-{\int }_b^{a}f\left(x\right)dx\\ 3.& {\int }_a^{b}f\left(x\right)dx={\int }_a^{c}f\left(x\right)dx+{\int }_c^{b}f\left(x\right)dx,\text{where}a<c<b\\ 4.& {\int }_0^{a}f\left(x\right)dx={\int }_0^{a}f(a-x)dx\\ 5.& {\int }_0^{\pi /2}{\sin }^{n}xdx={\int }_0^{\pi /2}{\sin }^n(\frac{\pi }{2}-x)dx={\int }_0^{\pi /2}{\cos }^{n}xdx\\ 6.& {\int }_{-a}^{a}f\left(x\right)dx=0,\text{when}f\left(x\right)\text{is an odd function}\\ 7.& {\int }_{-a}^{a}f\left(x\right)dx=2{\int }_0^{a}f\left(x\right)dx,\text{when}f\left(x\right)\text{is an even function}\\ 8.& {\int }_0^{2a}f\left(x\right)dx=2{\int }_0^{a}f\left(x\right)dx,\text{when}f(2a-x)=f\left(x\right)\\ 9.& {\int }_0^{2a}f\left(x\right)dx=0,\text{when}f(2a-x)=-f\left(x\right)\\ 10.& {\int }_0^{na}f\left(x\right)dx=n{\int }_0^{a}f\left(x\right)dx,\text{when}f\left(x\right)=f(a+x)\end{array}\end{array}$$

7 Beta and Gamma functions

Beta function	$\beta (m,n)={\int }_0^1{x}^{m-1}(1-x{)}^{n-1}dx$, where $m>0,n>0$
Gamma function	$\Gamma \left(n\right)={\int }_0^{\infty }{e}^{-x}{x}^{n-1}dx$, where $n>0$

8 Important properties and values of beta and gamma function

$\beta (m,n)$	$\beta (n,m)$
$\beta (m,n)$	$2{\int }_0^{\pi /2}{\sin }^{2m-1}\theta {\cos }^{2n-1}\theta d\theta$
$\beta (m,n)$	$\frac{\Gamma \left(m\right)\Gamma \left(n\right)}{\Gamma (m+n)}$
$\beta (m,n)$	${\int }_0^{\infty }\frac{x^{m-1}}{(1+x{)}^{m+n}}dx$
$\beta (m,n)$	${\int }_0^{\infty }\frac{x^{n-1}}{(1+x{)}^{m+n}}dx$
$\Gamma \left(1\right)$	$1$
$\Gamma (n+1)$	$n\Gamma \left(n\right)$
$\Gamma \left(m\right)\Gamma (1-m)$	$\frac{\pi }{\sin m\pi }$, $0<m<1$
$\Gamma \left(\frac{1}{2}\right)$	$\sqrt{\pi }$

9 Applied integration

9.1 Quadrature

9.1.1 Cartesian curves

• Area between curve $y=f\left(x\right)$, $x$-axis, ordinates $x=a$ and $x=b$ is given by ${\int }_a^{b}f\left(x\right)dx$.
• Area between two curves $y=f\left(x\right)$ and $y=\varphi \left(x\right)$, and the two ordinates $x=a$ and $x=b$ where $f\left(x\right)>\varphi \left(x\right)$ in the interval $[a,b]$ is ${\int }_a^b\left[f\right(x)-\varphi (x\left)\right]dx$.

9.1.2 Polar curves

• Area bounded by the curve $r=f\left(\theta \right)$ and two radii vectors $\theta =\alpha$ and $\theta =\beta$ is $\frac{1}{2}{\int }_{\alpha }^{\beta }{r}^{2}d\theta$.
• Area bounded by the two curves $r_1=f_1\left(\theta \right)$ and $r_2=f_2\left(\theta \right)$, and two radii vectors $\theta =\alpha$ and $\theta =\beta$ is $\frac{1}{2}{\int }_{\alpha }^{\beta }(r_2^2-r_1^2)d\theta$.

9.2 Rectification

9.2.1 Cartesian curves

• Length of arc of curve $y=f\left(x\right)$ between points A and B with abscissa $a$ and $b$ respectively is ${\int }_a^b\sqrt{1+(\frac{dy}{dx}{)}^2}dx$.

• Length of arc of curve $x=f\left(y\right)$ between points A and B with ordinates $a$ and $b$ respectively is ${\int }_a^b\sqrt{1+(\frac{dx}{dy}{)}^2}dy$.

9.2.2 Parametric equations

• Length of arc of a parametric equation of curve $x=f\left(t\right),y=\varphi \left(t\right)$ between two points for which $t=t_1$ and $t=t_2$ is ${\int }_{t_1}^{t_2}\sqrt{(\frac{dx}{dt}{)}^2+(\frac{dy}{dt}{)}^2}dt$.

9.2.3 Polar equations

• Length of arc of polar curve $r=f\left(\theta \right)$ between two points for which $\theta ={\theta }_1$ and $\theta ={\theta }_2$ is ${\int }_{{\theta }_1}^{{\theta }_2}\sqrt{r^2+(\frac{dr}{d\theta }{)}^2}d\theta$.

• Length of arc of polar curve $\theta =f\left(r\right)$ between two points for which $r=r_1$ and $r=r_2$ is ${\int }_{r_1}^{r_2}\sqrt{1+(\frac{rd\theta }{dr}{)}^2}dr$.

9.3 Volume of solid of revolution

9.3.1 Volume of solid of revolution: about $x$ or $y$ axis

• Volume of solid formed when the area bounded by the curve $y=f\left(x\right)$, $x$-axis, and the ordinates $x=a$ and $x=b$ revolves about the $x$-axis is

$${\int }_a^b\pi y^{2}dx$$

• Volume of solid formed when the area bounded by curve $x=f\left(y\right)$, $y$-axis, and the abscissa $y=a$ and $y=b$ revolves about the $x$-axis is

$${\int }_a^b\pi x^{2}dy$$

9.3.2 Volume of solid of revolution: about an axis parallel to $x$ or $y$ axis

• Volume of solid formed when the area bounded by the curve $y=f\left(x\right)$, $y=c$, and the ordinates $x=a$ and $x=b$ revolves about the axis $y=c$ parallel to $x$-axis is

$$\begin{array}{ccc}& {\int }_a^b\pi (y-c{)}^{2}dx& \text{or}\\ & {\int }_a^b\pi (c-y{)}^{2}dx& \end{array}$$

• Volume of solid formed when the area bounded by curve $x=f\left(y\right)$, $x=c$, and the abscissa $y=a$ and $y=b$ revolves about the axis $x=c$ parallel to $y$-axis is

$$\begin{array}{ccc}& {\int }_a^b\pi (x-c{)}^{2}dy& \text{or}\\ & {\int }_a^b\pi (c-x{)}^{2}dy& \end{array}$$

9.3.3 Volume using polar equation

• If the equation of the curve is given in polar form and the curve revolves about the initial line, volume can be obtained by changing $x=r\cos \theta$ and $y=r\sin \theta$

$$\begin{array}{cc}{\int }_a^b\pi y^{2}dx& =\pi {\int }_{\alpha }^{\beta }{r}^2{\sin }^2\theta \frac{d\left(r\cos \theta \right)}{d\theta }d\theta \end{array}$$

where $\alpha$ and $\beta$ are the values of $\theta$ which corresponds to the value $a$ and $b$ of $x$.

9.4 Surface of solid of revolution

9.4.1 Surface of solid of revolution: about $x$ or $y$ axis

For conceptual understanding of surface of revolution see excellent illustrations here.

• Surface of solid formed when the area bounded by the curve $y=f\left(x\right)$, $x$-axis, and the ordinates $x=a$ and $x=b$ revolves about the $x$-axis is

$$2\pi {\int }_a^{b}y\frac{ds}{dx}dx$$ where $s$ is the length of the curve between the point whose abscissa is $a$ and any other point $P(x,y)$. Also $\frac{ds}{dx}=\sqrt{1+{\left(\frac{dy}{dx}\right)}^2}$.

• Surface of solid formed when the area bounded by the curve $x=f\left(y\right)$, $y$-axis, and the abscissa $y=a$ and $y=b$ revolves about the $y$-axis is

$$2\pi {\int }_a^{b}x\frac{ds}{dy}dy$$ where $s$ is the length of the curve between the point whose ordinate is $a$ and any other point $P\left(f\right(y),y)$. Also $\frac{ds}{dx}=\sqrt{1+{\left(\frac{dx}{dy}\right)}^2}$.