If $f$ & $g$ are functions of $x$ such that $g^{\prime}(x)=f(x)$ then,

$\int f(x) dx = g(x) + c \Leftrightarrow \frac{d}{dx}{g(x) + c} = f(x)$

where $c$ is called the constant of integration.

Standard Formula:

• $\int(a x+b)^{n} d x=\frac{(a x+b)^{n+1}}{a(n+1)}+c, n \neq-1$

• $\int \frac{\mathrm{dx}}{\mathrm{ax}+\mathrm{b}}=\frac{1}{\mathrm{a}} \operatorname{\ell n}(\mathrm{ax}+\mathrm{b})+\mathrm{c}$

• $\int e^{a x+b} d x=\frac{1}{a} e^{a x+b}+c$

• $\int \mathrm{a}^{\mathrm{px+q}} \mathrm{dx}=\frac{1}{\mathrm{p}} \frac{\mathrm{a}^{\mathrm{px}+\mathrm{q}}}{\ln \mathrm{a}}+\mathrm{c} ; a>0$

• $\int \sin (a x+b) d x=-\frac{1}{a} \cos (a x+b)+c$

• $\int \cos (a x+b) d x=\frac{1}{a} \sin (a x+b)+c$

• $\quad \int \tan (a x+b) d x=\frac{1}{a} \ln \sec (a x+b)+c$

• $\int \cot (a x+b) d x=\frac{1}{a} \ln \sin (a x+b)+c$

• $\int \sec ^{2}(a x+b) d x=\frac{1}{a} \tan (a x+b)+c$

• $\int \operatorname{cosec}^{2}(a x+b) d x=-\frac{1}{a} \cot (a x+b)+c$

• $\int \sec x d x=\ell n (\sec x+\tan x)+c$

$ \ell n \tan \left(\frac{\pi}{4}+\frac{x}{2}\right)+c$

• $\int \operatorname{cosec} x d x=\ell n(\operatorname{cosec} x-\cot x)+c$

• $\quad \int \frac{d x}{\sqrt{a^{2}-x^{2}}}=\sin ^{-1} \frac{x}{a}+c$

• $\int \frac{\mathrm{dx}}{\mathrm{a}^{2}+\mathrm{x}^{2}}=\frac{1}{\mathrm{a}} \tan ^{-1} \frac{\mathrm{x}}{\mathrm{a}}+\mathrm{c}$

• $\int \frac{d x}{|x| \sqrt{x^{2}-a^{2}}}=\frac{1}{a} \sec ^{-1} \frac{x}{a}+c$

• $\int \frac{\mathrm{dx}}{\sqrt{\mathrm{x}^{2}+\mathrm{a}^{2}}}=\operatorname{\ell n}\left[\mathrm{x}+\sqrt{\mathrm{x}^{2}+\mathrm{a}^{2}}\right]+\mathrm{c}$

• $\int \frac{\mathrm{dx}}{\sqrt{\mathrm{x}^{2}-\mathrm{a}^{2}}}=\ell \mathrm{n}\left[\mathrm{x}+\sqrt{\mathrm{x}^{2}-\mathrm{a}^{2}}\right]+\mathrm{c}$

• $\quad \int \frac{\mathrm{dx}}{\mathrm{a}^{2}-\mathrm{x}^{2}}=\frac{1}{2 \mathrm{a}} \operatorname{\ell n}\left|\frac{\mathrm{a}+\mathrm{x}}{\mathrm{a}-\mathrm{x}}\right|+\mathrm{c}$

• $\int \frac{\mathrm{dx}}{\mathrm{x}^{2}-\mathrm{a}^{2}}=\frac{1}{2 \mathrm{a}} \ell \mathrm{n}\left|\frac{\mathrm{x}-\mathrm{a}}{\mathrm{x}+\mathrm{a}}\right|+\mathrm{c}$

• $\int \sqrt{a^{2}-x^{2}} d x=\frac{x}{2} \sqrt{a^{2}-x^{2}}+\frac{a^{2}}{2} \sin ^{-1} \frac{x}{a}+c$

• $\int \sqrt{\mathrm{x}^{2}+\mathrm{a}^{2}} \mathrm{dx}=\frac{\mathrm{x}}{2} \sqrt{\mathrm{x}^{2}+\mathrm{a}^{2}}+\frac{\mathrm{a}^{2}}{2} \ln (\mathrm{x}+\sqrt{\mathrm{x}^{2}+\mathrm{a}^{2}})+\mathrm{c}$

• $\int \sqrt{\mathrm{x}^{2}-\mathrm{a}^{2}} \mathrm{dx}=\frac{\mathrm{x}}{2} \sqrt{\mathrm{x}^{2}-\mathrm{a}^{2}}-\frac{\mathrm{a}^{2}}{2} \ln (\mathrm{x}+\sqrt{\mathrm{x}^{2}-\mathrm{a}^{2}})+\mathrm{c}$

Integration by Subsitutions

$\quad $ If we subsitute $f(x)=t$, then $f^{\prime}(x) d x=d t$

Integration by Part :

$\int(f(x) g(x)) d x=f(x) \int(g(x)) d x-\int (\frac{d}{d x}(f(x)) \int(g(x)) d x) d x$

Integration of types

$ \int \frac{dx}{a x^{2}+b x+c} , \int \frac{dx}{\sqrt{a x^{2}+b x+c}} ,\int \sqrt{a x^{2}+b x+c} d x $

$\quad $ Make the substitution $x+\frac{b}{2 a}=t$

Integration of type

$ \int \frac{p x+q}{a x^{2}+b x+c} d x, \int \frac{p x+q}{\sqrt{a x^{2}+b x+c}} d x,\int(p x+q) \sqrt{a x^{2}+b x+c} d x $

$\quad $ Make the substitution $x+\frac{b}{2 a}=t$

$\quad $ then split the integral as some of two integrals one containing the linear term and the other containing constant term.

Integration of trigonometric functions

• $\int \frac{d x}{a+b \sin ^{2} x}$

$\int \frac{d x}{a+b \cos ^{2} x}$

$\int \frac{d x}{a \sin ^{2} x+b \sin x \cos x+c \cos ^{2} x}$

$\quad $ put $\tan x=t$

• $\int \frac{d x}{a+b \sin x}$

$\int \frac{d x}{a+b \cos x}$

$\int \frac{d x}{a+b \sin x+c \cos x}$

$\quad $ put $\tan \frac{x}{2}=t$

• $\int \frac{a \cdot \cos x+b \cdot \sin x+c}{\ell \cdot \cos x+m \cdot \sin x+n} d x$

$\quad $ Express $N r \equiv A(D r)+B \frac{d}{d x}(D r)+c$

Integration of type:

$\int \frac{\mathrm{x}^{2} \pm 1}{\mathrm{x}^{4}+K \mathrm{x}^{2}+1} \mathrm{dx}$

$\quad $ where ${K}$ is any constant.

$\quad $ Divide $Nr$ and $Dr$ by $x^2$ and put $x \mp \frac{1}{x} = t$.

Integration of type:

$\int \frac{d x}{(a x+b) \sqrt{p x+q}}$

$\int \frac{d x}{\left(a x^{2}+b x+c\right) \sqrt{p x+q}} \quad \text { put }\quad p x+q=t^{2}$

Integration of type:

$ \int \frac{d x}{(a x+b) \sqrt{p x^{2}+q x+r}} \quad \text { put } \quad x+b=\frac{1}{t} $

$ \int \frac{d x}{\left(a x^{2}+b\right) \sqrt{p x^{2}+q}} \quad\text { put }\quad x=\frac{1}{t} $

Integration of type:

• $\int \sqrt{\frac{x-\alpha}{\beta-x}} d x $

$ \int \sqrt{(x-\alpha)(\beta-x)} \quad\text { put }\quad x=\alpha \cos ^2 \theta+\beta \sin ^2 \theta $

• $\int \sqrt{\frac{x-\alpha}{x-\beta}} d x \quad \text { put }\quad x=\alpha \cos ^2 \theta+\beta \sin ^2 \theta $

• $ \int \sqrt{(x-\alpha)(x-\beta)} \quad\text { put }\quad x=\alpha \sec ^2 \theta-\beta \tan ^2 \theta$

• $\int \frac{d x}{\sqrt{(x-\alpha)(x-\beta)}} \quad \text { put }\quad \mathrm{x}-\alpha=\mathrm{t}^2 \text { or } \mathrm{x}-\beta=\mathrm{t}^2 $

Theorems on integration:

• $ \int c f(x) d x=c \int f(x) d x$

• $ \int[f(x) \pm g(x)] d x=\int f(x) d x \pm \int g(x) d x$

• $ \int f(x) d x=g(x)+c$

• $ \Rightarrow \int f(a x+b) d x=\frac{F(a x+b)}{a}+c, \quad a\ne0 $

Integration of type $\int \sin m x \cdot \cos n x d x$ :

$\quad$ Case 1. If $m$ and $n$ are even natural numbers then express $\sin m x \cos n x$ in the terms of sines and cosines of multiples of $x$ by using trigonometric results or De’ Moivere’s theorem.

$\quad$ Case 2.

• If $m$ is an odd natural number then put $\cos x=t$.
• If $\mathrm{n}$ is an odd natural number then put $\sin \mathrm{x}=\mathrm{t}$.
• If both $\mathrm{m}$ and $\mathrm{n}$ are odd natural numbers then put either $\sin \mathrm{x}=\mathrm{t}$ or $\cos \mathrm{x}=\mathrm{t}$.

$\quad$ Case 3. When $m+n$ is a negative even integer then put tan $x=t$.

Reduction formula of $\int \tan ^n x d x, \int \cot ^n x d x, \int \sec ^n x d x, \int \operatorname{cosec}^n x d x , n\ne1$:

• If $I_n =\int \tan ^n x d x$, then $ I_n =\frac{\tan ^{n-1} x}{n-1}-I_{n-2} $

• If $I_n =\int \cot ^n x d x$, then

$ I_n=\frac{\cot ^{n-1} x}{n-1}-I_{n-2} $

• If $I_n =\int \sec ^n x d x$, then

$ I_n=\frac{\tan x \sec ^{n-2} x}{n-1}+\frac{n-2}{n-1} I_{n-2} $

• If $I_n =\int \operatorname{cosec}^n x$, then

$I_n=\frac{\cot x \operatorname{cosec}^{n-2} x}{n-1}+\frac{n-2}{n-1} I_{n-2}$