Important Formulas For Integration.

$\displaystyle \bullet \;\; \int x^{n} d x=\frac{x^{n+1}}{n+1}+c \quad \text{ where}\;\; n \neq-1$

$\displaystyle \bullet \;\; \int \cos x d x=\sin x+c$

$\displaystyle \bullet \;\; \int \sin x d x=-\cos x+c$

$\displaystyle \bullet \;\; \int \sec x \tan x d x=\sec x+c$

$\displaystyle \bullet \;\; \int \sec ^{2} x d x=\tan x+c$

$\displaystyle \bullet \;\; \int \csc ^{2} x d x=-\cot x+c$

$\displaystyle \bullet \;\; \int \csc x \cot x d x=-\csc x+c$

$\displaystyle \bullet \;\; \int \frac{d x}{\sqrt{1-x^{2}}}=\sin ^{-1} x+c$

$\displaystyle \bullet \;\; \int \frac{d x}{\sqrt{1-x^{2}}}=-\cos ^{-1} x+c$

$\displaystyle \bullet \;\; \int \frac{d x}{1+x^{2}}=\tan ^{-1} x+c$

$\displaystyle \bullet \;\; \int \frac{d x}{x \sqrt{x^{2}-1}}=-\csc ^{-1} x+c$

$\displaystyle \bullet \;\; \int \frac{d x}{1+x^{2}}=-\cot ^{-1} x+c$

$\displaystyle \bullet \;\; \int \frac{d x}{x \sqrt{x^{2}-1}}=\sec ^{-1} x+c$

$\displaystyle \bullet \;\; \int a^{x} d x=\frac{a^{x}}{\ln a}+c$

$\displaystyle \bullet \;\; \int e^{a x} d x=\frac{e^{a} x}{a}+c$

$\displaystyle \bullet \;\; \int \frac{1}{x} d x=\ln |x|+c$

$\displaystyle \bullet \;\; \int \tan x d x=\ln |\sec x|+c$

$\displaystyle \bullet \;\; \int \cot x d x=\ln |\sin x|+c$

$\displaystyle \bullet \;\; \int \sec x d x=\ln |\sec x+\tan x|+c$

$\displaystyle \bullet \;\; \int \csc x d x=\ln |\csc x-\cot x|+c$

$\displaystyle \bullet \;\; \int \frac{d x}{x^{2}-a^{2}}=\frac{1}{2 a} \ln \left|\frac{x-a}{x+a}\right|+c$

$\displaystyle \bullet \;\; \int \frac{d x}{a^{2}-x^{2}}=\frac{1}{2 a} \ln \left|\frac{a+x}{a-x}\right|+c$

$\displaystyle \bullet \;\; \int \frac{d x}{a^{2}+x^{2}}=\frac{1}{a} \tan ^{-1} \frac{x}{a}+c$

$\displaystyle \bullet \;\; \int \frac{d x}{\sqrt{x^{2}-a^{2}}}=\ln \left|x+\sqrt{x^{2}-a^{2}}\right|+c$

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

$\displaystyle \bullet \;\; \int \frac{d x}{\sqrt{x^{2}+a^{2}}}=\ln \left|x+\sqrt{x^{2}+a^{2}}\right|+c$

$\displaystyle \bullet \;\; \int \sqrt{x^{2}-a^{2}} d x=\frac{x}{2} \sqrt{x^{2}-a^{2}}-\frac{a^{2}}{2} \ln \left|x+\sqrt{x^{2}-a^{2}}\right|+c$

$\displaystyle \bullet \;\; \int \sqrt{x^{2}+a^{2}} d x=\frac{x}{2} \sqrt{x^{2}+a^{2}}+\frac{a^{2}}{2} \ln \left|x+\sqrt{x^{2}+a^{2}}\right|+c$

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

$\displaystyle \bullet \;\; \int \arcsin x d x=x \arcsin x+\sqrt{1-x^{2}}+C$

$\displaystyle \bullet \;\; \int \arccos x \mathrm{dx}=\mathrm{x} \arccos x-\sqrt{1-\mathrm{x}^{2}}+\mathrm{C}$

$\displaystyle \bullet \;\;\int \arctan x \mathrm{dx}=\mathrm{x} \arctan \mathrm{x}-\frac{1}{2} \ln \left(\mathrm{x}^{2}+1\right)+\mathrm{C}$

$\displaystyle \bullet \;\;\int \operatorname{arccot} x d x=x \operatorname{arccot} x+\frac{1}{2} \ln \left(x^{2}+1\right)+C$

Integrals of Hyperbolic Functions

$\displaystyle \bullet \;\;\int \sinh x d x=\cosh x+C$

$\displaystyle \bullet \;\;\int \cosh x d x=\sinh x+C$

$\displaystyle \bullet \;\;\int \tanh x d x=\ln \cosh x+C$

$\displaystyle \bullet \;\;\int \operatorname{coth} x d x=\ln |\sinh x|+C$

$\displaystyle \bullet \;\;\int \operatorname{sech}^{2} x d x=\tanh x+C$

$\displaystyle \bullet \;\;\int \operatorname{csch}^{2} x d x=-\operatorname{coth} x+C$

$\displaystyle \bullet \;\;\int \operatorname{sech} x \tanh x d x=-\operatorname{sech} x+C$

$\displaystyle \bullet \;\;\int x^{n} \ln ^{m} x d x=\frac{x^{n+1} \ln ^{m} x}{n+1}-\frac{m}{n+1} \int x^{n} \ln ^{m-1} x d x$

$\displaystyle \bullet \;\;\int \frac{\ln ^{\mathrm{m}} \mathrm{x}}{\mathrm{x}^{\mathrm{n}}} \mathrm{dx}=-\frac{\ln ^{\mathrm{m}} \mathrm{x}}{(\mathrm{n}-1) \mathrm{x}^{\mathrm{n}-1}}+\frac{\mathrm{m}}{\mathrm{n}-1} \int \frac{\ln ^{\mathrm{m}-1} \mathrm{x}}{\mathrm{x}^{\mathrm{n}}} \mathrm{dx}, \mathrm{n} \neq 1$

$\displaystyle \bullet \;\;\int \ln ^{\mathrm{n}} \mathrm{xdx}=\mathrm{x} \ln ^{\mathrm{n}} \mathrm{x}-\mathrm{n} \int \ln ^{\mathrm{n}-1} \mathrm{xdx}$

$\displaystyle \bullet \;\;\int x^{m} \sinh x d x=x^{m} \cosh x-m \int x^{m-1} \cosh x d x$

$\displaystyle \bullet \;\;\int x^{m} \cosh x d x=x^{m} \sinh x-m \int x^{m-1} \sinh x d x$

$\displaystyle \bullet \;\;\int x^{m} \sin x d x=-x^{m} \cos x+m \int x^{m-1} \cos x d x$

$\displaystyle \bullet \;\;\int x^{m} \cos x d x=x^{m} \sin x-m \int x^{m-1} \sin x d x$



INTEGRALS BY PARTIAL FRACTIONS

 $\displaystyle \bullet \;\; \int \frac{p x+q}{(x-a)(x-b)}=\frac{A}{x-a}+\frac{B}{x-b}\;\;$ such that $\;\;a \neq b$

$\displaystyle \bullet \;\; \int \frac{p x+q}{(x-a)^{2}}=\frac{A}{x-a}+\frac{B}{(x-a)^{2}}$

$\displaystyle \bullet \;\; \int \frac{p x^{2}+q x+r}{(x-a)(x-b)(x-c)}=\frac{A}{x-a}+\frac{B}{(x-a)}+\frac{C}{x-c}$

$\displaystyle \bullet \;\; \int \frac{p x^{2}+q x+r}{(x-a)^{2}(x-b)}=\frac{A}{x-a}+\frac{B}{(x-a)^{2}}+\frac{C}{x-b}$

$\displaystyle \bullet \;\; \int \frac{p x^{2}+q x+r}{(x-a)\left(x^{2}+b x+c\right)}=\frac{A}{x-a}+\frac{B x+C}{\left(x^{2}+b x+c\right)}$

NOTE - Here Given $x^{2}+b x+c$ doesn't have any further linear factors

INTEGRATION BY PARTS FORMULAS

$\displaystyle \int f(x) g(x) d x=f(x) \int g(x) d x-\int\left(f^{\prime}(x) \int g(x) d x\right) d x$

RULE-Here we have to decide which is first function for that here is a rule named as ILATE Where

$I \rightarrow\;\;$ Inverse trigonemtry functions

$\mathrm{L} \rightarrow$ Logarithmic functions

$A \rightarrow$ Arithmatic functions

$\mathrm{T} \rightarrow$ Trigonometric functions

$\mathrm{E} \rightarrow$ Exponential functions

FORMULAS FOR DEFINITE INTEGRATION

$\displaystyle \bullet \;\; \int_{\alpha}^{\beta} f(x) d x=\int_{\alpha}^{\beta} f(t) d t$

Here it explain that we can change the variable

$\displaystyle \bullet \;\; \int_{\alpha}^{b} f(x) d x=-\int_{b}^{\alpha} f(x) d x$

In above this tells us that you can flip limits but integral will be negative value of original one.

$\displaystyle \bullet \;\; \int_{\alpha}^{\beta} f(x) d x=\int_{\alpha}^{\gamma} f(x) d x+\int_{\gamma}^{\beta} f(x) d x$

In above it tells us that we can break the limits.

 $\displaystyle 49.)\;\; \int_{\alpha}^{\beta} f(x) d x=\int_{\alpha}^{\beta} f(\alpha+\beta-x) d x$

$\displaystyle \bullet \;\; \int_{0}^{\alpha} f(x) d x=\int_{0}^{\alpha} f(\alpha-x) d x$

$\displaystyle \bullet \;\; \int_{0}^{2 \alpha} f(x) d x=\int_{0}^{\alpha} f(x) d x+\int_{0}^{\alpha} f(2 \alpha-x) d x$

$\displaystyle \bullet \;\; \int_{0}^{2 \alpha} f(x) d x= \begin{cases}2 \displaystyle \int_{0}^{\alpha} f(x) d x & \text { if } f(2 \alpha-x)=f(x) \\ 0 & \text { if } f(2 \alpha-x)=-f(x)\end{cases}$

$\displaystyle \bullet \;\; \int_{-\alpha}^{\alpha} f(x) d x= \begin{cases}2 \displaystyle \int_{0}^{\alpha} f(x) d x & \text { if } f(-x)=f(x) \\ 0 & \text { if } f(-x)=-f(x)\end{cases}$

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