Definite integral formula
\begin{aligned} &\int_a^b f(x) \, \mathrm{d}x=F(b)-F(a) \ ; \text{where} \ \int f(x) \, \mathrm{d}x=F(x)+c \\ &\int_a^b f(x) \, \mathrm{d}x=-\int_b^a f(x) \, \mathrm{d}x \\ &\int_a^b f(x) \, \mathrm{d}x=\int_a^b f(a+b-x) \, \mathrm{d}x \\ &\int_{-a}^a f(x) \, \mathrm{d}x=2\int_0^a f(x)\, \mathrm{d}x\ ; \text{where} \ f(x) \ \text{is an even function} \\ &\int_{-a}^a f(x) \, \mathrm{d}x=0 \ ; \text{where} \ f(x) \ \text{is an odd function} \\ \end{aligned}
The integral of a sum is the sum of the integrals:
\int f(x)+g(x)\;dx=\int f(x)\;dx+\int g(x)\;dx
likewise, constants ‘go through’ the integral sign:
\int c\cdot f(x)\;dx=c\cdot \int f(x)\;dx
List of Basic Integral Formulas
\int x^n\; dx = {1\over n+1}x^{n+1}+C
\hbox{ unless n=-1 }
\int e^x \;dx = e^x+C
\int {1\over x} \;dx= \ln x+C
\int \sin x\;dx=-\cos x+C
\int \cos x\;dx= \sin x + C
\int \sec^2 x\;dx=\tan x+C
\int {1\over 1+x^2} \; dx=\arctan x+C
\int a^x \;dx= {a^x\over \ln a}+C
\int \log_a x\;dx={1\over \ln a}\cdot{1\over x}+C
\int { 1 \over \sqrt{1-x^2 }} \; dx=\arcsin x+C
\int { 1 \over x\sqrt{x^2-1 }} \; dx=\hbox{ arcsec}\, x+C
Fundamental Integration Formulas chart
\begin{aligned} &\int x^n\, \mathrm{d} x=\frac{x^{n+1}}{n+1}+c \\ &\int e^x \, \mathrm{d}x=e^x+c \\ &\int a^x \, \mathrm{d}x=\frac{a^x}{\log_e a}+c \\ &\int \frac{1}{x} \, \mathrm{d}x=\log_e x +c \\ &\int k\, \mathrm{d} x=k x+c \end{aligned}
Trigonometric Integration formulas Chart
\begin{aligned} &\int \sin x \, \mathrm{d} x=-\cos x+c \\&\int \cos x \, \mathrm{d} x=\sin x+c \\&\int \tan x \, \mathrm{d} x=\ln |\sec x|+c \\&\int \sec x \, \mathrm{d} x=\ln |\tan x+\sec x|+c \\&\int \sin ^2 x \, \mathrm{d} x=\frac{1}{2}(x-\sin x \cos x)+c \\&\int \cos ^2 x \, \mathrm{d} x=\frac{1}{2}(x+\sin x \cos x)+c \\&\int \tan ^2 x \, \mathrm{d} x=\tan x-x+c \\&\int \sec ^2 x \, \mathrm{d} x=\tan x+c \\ &\int \sin x \cos ^n x \, \mathrm{d} x=-\frac{\cos ^{n+1} x}{n+1}+c \\ &\int \sin ^n x \cos x \, \mathrm{d} x=\frac{\sin ^{n+1} x}{n+1} +c\\&\int \sinh x \, \mathrm{d}x=\cosh x +c \\ &\int \cosh x \, \mathrm{d}x=\sinh x +c\end{aligned}
Inverse Trigonometric Integration Chart
\begin{aligned} &\int \sin^{-1} x \, \mathrm{d} x=x \sin^{-1} x+\sqrt{1-x^2}+c \\&\int \cos^{-1} x \, \mathrm{d} x=x \cos^{-1} x-\sqrt{1-x^2}+c \\&\int \tan^{-1} x \, \mathrm{d} x=x \tan^{-1} x-\frac{1}{2} \ln \left(x^2+1\right) +c\\&\int \cot^{-1} x \, \mathrm{d} x=x \cot^{-1} x+\frac{1}{2} \ln \left(x^2+1\right) +c\\ &\int \csc ^{-1} x \, \mathrm{d} x=x \csc ^{-1} x+\ln \left|x+\sqrt{x^2-1}\right|+c \\ &\int \sec ^{-1} x \, \mathrm{d} x=x \sec ^{-1} x-\ln \left|x+\sqrt{x^2-1}\right|+c \end{aligned}
Other Integration formulas chart
\begin{aligned} &\int e^{c x} \sin b x \, \mathrm{d} x=\frac{e^{c x}}{c^2+b^2}(c \sin b x-b \cos b x) +c\\ &\int e^{c x} \cos b x \, \mathrm{d} x=\frac{e^{c x}}{c^2+b^2}(c \cos b x+b \sin b x) +c\\&\int \frac{1}{x^2+a^2} \, \mathrm{d} x=\frac{1}{a} \tan^{-1} \frac{x}{a} +c\\&\int \frac{1}{x^2-a^2} \, \mathrm{d} x= \left\{\begin{array}{l} \displaystyle \frac{1}{2 a} \ln \left( \frac{a-x}{a+x} \right) +c\\\displaystyle \frac{1}{2 a} \ln \left( \frac{x-a}{x+a} \right)+c \end{array}\right. \end{aligned}