Standard integrals

These formulas should be remembered for solving integral problems:

\(\int \! c f(x) \,\mathrm{d}x = c\int \! f(x) \,\mathrm{d}x\), where \(c\) is a constant
\(\int \! \{f(_1)x \pm f_2(x)\} \,\mathrm{d}x = \int \! f_1(x) \,\mathrm{d}x \pm \int \! f_2(x) \,\mathrm{d}x\)
\(\int \! x^n \,\mathrm{d}x=\frac{x^{n+1}}{n+1}\), where \(n \neq -1\)
\(\int \! \frac{1}{x} \,\mathrm{d}x=\log x\)
\(\int \! \,\mathrm{d}x=x\)
\(\int \! e^x \,\mathrm{d}x=e^x\)
\(\int \! e^{ax} \,\mathrm{d}x=\frac{1}{a}e^{ax}\)
\(\int \! a^x \,\mathrm{d}x=\frac{a^x}{\log a}\), where \(a>0\)
\(\int \! \sin mx \,\mathrm{d}x=-\frac{\cos mx}{m}\)
\(\int \! \cos mx \,\mathrm{d}x=\frac{\sin mx}{m}\)
\(\int \! \sec^2 x \,\mathrm{d}x=\tan x\)
\(\int \! \csc^2 x \,\mathrm{d}x=-\cot x\)
\(\int \! \sec x \tan x \,\mathrm{d}x=\sec x\)
\(\int \! \csc x \cot x \,\mathrm{d}x=-\csc x\)
\(\int \! \sin \text{h}x \,\mathrm{d}x=\cos \text{h}x\)
\(\int \! \cos \text{h}x \,\mathrm{d}x=\sin \text{h}x\)
\(\int \! \sec \text{h}^2x \,\mathrm{d}x=\tan \text{h}x\)
\(\int \! \csc \text{h}^2x \,\mathrm{d}x=-\cot \text{h}x\)
\(\int \! \sec \text{h}x \tan \text{h}x \,\mathrm{d}x=-\sec \text{h}x\)
\(\int \! \csc \text{h}x \cot \text{h}x \,\mathrm{d}x=-\csc \text{h}x\)
\(\int \! \frac{f'(x)}{f(x)} \,\mathrm{d}x=\log(f(x))\)
\(\int \! \tan x \,\mathrm{d}x=\log(\sec x)\)
\(\int \! \cot x \,\mathrm{d}x=\log(\sin x)\)
\(\int \! \csc x \,\mathrm{d}x=\log(\tan \frac{x}{2})=\log(\csc x - \cot x)\)
\(\int \! \sec x \,\mathrm{d}x=\log\{\tan\left(\frac{\pi}{4}+\frac{x}{2}\right)\}=\log(\sec x + \tan x)\)
\(\int \! \frac{\mathrm{d}x}{x^2 + a^2}=\frac{1}{a}\tan^{-1}\frac{x}{a}\)
\(\int \! \frac{\mathrm{d}x}{x^2 - a^2}=\frac{1}{2a}\log \frac{x-a}{x+a}\), where \(x>a\)
\(\int \! \frac{\mathrm{d}x}{a^2 - x^2}=\frac{1}{2a}\log\frac{a+x}{a-x}\), where \(x<a\)
\(\int \! \frac{\mathrm{d}x}{\sqrt{a^2 - x^2}}=\sin^{-1}\frac{x}{a}\)
\(\int \! \frac{\mathrm{d}x}{\sqrt{x^2 - a^2}}=\log(x + \sqrt{x^2 - a^2})=\cos \text{h}^{-1}\frac{x}{a}\)
\(\int \! \frac{\mathrm{d}x}{\sqrt{x^2 + a^2}}=\log(x + \sqrt{x^2 + a^2})=\sin \text{h}^{-1}\frac{x}{a}\)
\(\int \! \frac{\mathrm{d}x}{x\sqrt{x^2 - a^2}}=\frac{1}{a}\sec^{-1}\frac{x}{a}\)
\(\int \! uv \,\mathrm{d}x=u\int \! v \,\mathrm{d}x-\int \! (\frac{du}{dx}.\int \! v \,\mathrm{d}x) \,\mathrm{d}x\)
\(\int \! e^{ax}\cos bx \,\mathrm{d}x=\frac{e^{ax}(a\cos bx + b \sin bx)}{a^2 + b^2}=\frac{e^{ax}}{\sqrt{a^2 + b^2}}\cos(bx - \tan^{-1}\frac{b}{a})\)
\(\int \! e^{ax}\sin bx \,\mathrm{d}x=\frac{e^{ax}(a\sin bx - b\cos bx)}{a^2 + b^2}\)
\(\int \! \sqrt{x^2 + a^2} \,\mathrm{d}x=\frac{x\sqrt{x^2 + a^2}}{2} + \frac{a^2}{2}\log(x + \sqrt{x^2 + a^2})\\=\frac{x\sqrt{x^2 + a^2}}{2} + \frac{a^2}{2}\sin \text{h}^{-1}\frac{x}{a}\)
\(\int \! \sqrt{x^2 - a^2} \,\mathrm{d}x=\frac{x\sqrt{x^2 - a^2}}{2} - \frac{a^2}{2}\log(x + \sqrt{x^2 - a^2})\\=\frac{x\sqrt{x^2 - a^2}}{2} - \frac{a^2}{2}\cos \text{h}^{-1}\frac{x}{a}\)
\(\int \! \sqrt{a^2 - x^2} \,\mathrm{d}x=\frac{x\sqrt{a^2 - x^2}}{2} + \frac{a^2}{2}\sin^{-1}\frac{x}{a}\)
\(\int \! e^x \{f(x) + f'(x) \,\mathrm{d}x\}=e^x f(x)\)

5 Methods of integration

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

6 Properties of definite integrals

\[\begin{align*} \begin{split} 1. & \int_{a}^{b} \! f(x) \,\mathrm{d}x=\int_{a}^{b} \! f(t) \,\mathrm{d}t\\ 2. & \int_{a}^{b} \! f(x) \,\mathrm{d}x=-\int_{b}^{a} \! f(x) \,\mathrm{d}x\\ 3. & \int_{a}^{b} \! f(x) \,\mathrm{d}x=\int_{a}^{c} \! f(x) \,\mathrm{d}x + \int_{c}^{b} \! f(x) \,\mathrm{d}x, \text{where } a<c<b\\ 4. & \int_{0}^{a} \! f(x) \,\mathrm{d}x=\int_{0}^{a} \! f(a-x) \,\mathrm{d}x\\ 5. & \int_{0}^{\pi/2} \! \sin^nx \,\mathrm{d}x=\int_{0}^{\pi/2} \! \sin^n\left(\frac{\pi}{2}-x\right) \,\mathrm{d}x=\int_{0}^{\pi/2} \! \cos^nx \,\mathrm{d}x\\ 6. & \int_{-a}^{a} \! f(x) \,\mathrm{d}x=0, \text{when } f(x) \text{ is an odd function}\\ 7. & \int_{-a}^{a} \! f(x) \,\mathrm{d}x=2\int_{0}^{a} \! f(x) \,\mathrm{d}x, \text{when } f(x) \text{is an even function}\\ 8. & \int_{0}^{2a} \! f(x) \,\mathrm{d}x=2\int_{0}^{a} \! f(x) \,\mathrm{d}x, \text{when } f(2a-x)=f(x)\\ 9. & \int_{0}^{2a} \! f(x) \,\mathrm{d}x=0, \text{when } f(2a-x)=-f(x)\\ 10. & \int_{0}^{na} \! f(x) \,\mathrm{d}x=n\int_{0}^{a} \! f(x) \,\mathrm{d}x, \text{when } f(x)=f(a+x)\\ \end{split} \end{align*}\]

7 Beta and Gamma functions

Beta function	\(\beta(m,n)=\int_{0}^{1} \! x^{m-1} (1-x)^{n-1} \,\mathrm{d}x\), where \(m>0, n>0\)
Gamma function	\(\Gamma(n)=\int_{0}^{\infty} \! e^{-x}x^{n-1} \,\mathrm{d}x\), where \(n>0\)

8 Important properties and values of beta and gamma function

\(\beta(m,n)\)	\(\beta(n,m)\)
\(\beta(m,n)\)	\(2\int_{0}^{\pi/2} \! \sin^{2m-1} \theta \cos^{2n-1} \theta \,\mathrm{d}\theta\)
\(\beta(m,n)\)	\(\frac{\Gamma(m)\Gamma(n)}{\Gamma(m+n)}\)
\(\beta(m,n)\)	\(\int_{0}^{\infty} \! \frac{x^{m-1}}{(1+x)^{m+n}} \,\mathrm{d}x\)
\(\beta(m,n)\)	\(\int_{0}^{\infty} \! \frac{x^{n-1}}{(1+x)^{m+n}} \,\mathrm{d}x\)
\(\Gamma(1)\)	\(1\)
\(\Gamma(n+1)\)	\(n\Gamma(n)\)
\(\Gamma(m)\Gamma(1-m)\)	\(\frac{\pi}{\sin m\pi}\), \(0<m<1\)
\(\Gamma(\frac{1}{2})\)	\(\sqrt{\pi}\)

9 Applied integration

9.1 Quadrature

9.1.1 Cartesian curves

Area between curve \(y=f(x)\), \(x\)-axis, ordinates \(x=a\) and \(x=b\) is given by \(\int_{a}^{b} \! f(x) \,\mathrm{d}x\).
Area between two curves \(y=f(x)\) and \(y=\phi(x)\), and the two ordinates \(x=a\) and \(x=b\) where \(f(x) > \phi(x)\) in the interval \([a,b]\) is \(\int_{a}^{b} \! [f(x)-\phi(x)] \,\mathrm{d}x\).

9.1.2 Polar curves

Area bounded by the curve \(r=f(\theta)\) and two radii vectors \(\theta=\alpha\) and \(\theta=\beta\) is \(\frac{1}{2}\int_{\alpha}^{\beta} \! r^2 \,\mathrm{d}\theta\).
Area bounded by the two curves \(r_1=f_1(\theta)\) and \(r_2=f_2(\theta)\), and two radii vectors \(\theta=\alpha\) and \(\theta=\beta\) is \(\frac{1}{2}\int_{\alpha}^{\beta} \! (r_2^2-r_1^2) \,\mathrm{d}\theta\).

9.2 Rectification

9.2.1 Cartesian curves

Length of arc of curve \(y=f(x)\) between points A and B with abscissa \(a\) and \(b\) respectively is \(\int_{a}^{b} \! \sqrt{1+(\frac{dy}{dx})^2} \,\mathrm{d}x\).
Length of arc of curve \(x=f(y)\) between points A and B with ordinates \(a\) and \(b\) respectively is \(\int_{a}^{b} \! \sqrt{1+(\frac{dx}{dy})^2} \,\mathrm{d}y\).

9.2.2 Parametric equations

Length of arc of a parametric equation of curve \(x=f(t), y=\phi(t)\) between two points for which \(t=t_1\) and \(t=t_2\) is \(\int_{t_1}^{t_2} \! \sqrt{(\frac{dx}{dt})^2+(\frac{dy}{dt})^2} \,\mathrm{d}t\).

9.2.3 Polar equations

Length of arc of polar curve \(r=f(\theta)\) between two points for which \(\theta=\theta_1\) and \(\theta=\theta_2\) is \(\int_{\theta_1}^{\theta_2} \! \sqrt{r^2 + (\frac{dr}{d\theta})^2} \,\mathrm{d}\theta\).
Length of arc of polar curve \(\theta=f(r)\) between two points for which \(r=r_1\) and \(r=r_2\) is \(\int_{r_1}^{r_2} \! \sqrt{1 + (\frac{r d\theta}{dr})^2} \,\mathrm{d}r\).

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 \(y=f(x)\), \(x\)-axis, and the ordinates \(x=a\) and \(x=b\) revolves about the \(x\)-axis is

\[ \int_{a}^{b} \! \pi y^2 \,\mathrm{d}x \]

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

\[ \int_{a}^{b} \! \pi x^2 \,\mathrm{d}y \]

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 \(y=f(x)\), \(y=c\), and the ordinates \(x=a\) and \(x=b\) revolves about the axis \(y=c\) parallel to \(x\)-axis is

\[\begin{equation*} \begin{split} & \int_{a}^{b} \! \pi (y-c)^2 \,\mathrm{d}x & \text{ or} \\ & \int_{a}^{b} \! \pi (c-y)^2 \,\mathrm{d}x \\ \end{split} \end{equation*}\]

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

\[\begin{equation*} \begin{split} & \int_{a}^{b} \! \pi (x-c)^2 \,\mathrm{d}y & \text{ or} \\ & \int_{a}^{b} \! \pi (c-x)^2 \,\mathrm{d}y \\ \end{split} \end{equation*}\]

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 \(y=r\sin \theta\)

\[\begin{equation*} \begin{split} \int_{a}^{b} \! \pi y^2 \,\mathrm{d}x &= \pi \int_{\alpha}^{\beta} \! r^2 \sin^2 \theta \frac{d(r\cos\theta)}{d\theta}\,\mathrm{d}\theta \end{split} \end{equation*}\]

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 \(y=f(x)\), \(x\)-axis, and the ordinates \(x=a\) and \(x=b\) revolves about the \(x\)-axis is

\[ 2 \pi \int_{a}^{b} \! y \frac{ds}{dx} \,\mathrm{d}x \] 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 \(x=f(y)\), \(y\)-axis, and the abscissa \(y=a\) and \(y=b\) revolves about the \(y\)-axis is

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