are used to solve calculus-related problems. In this article I have provided you with a complete Integration Formula Sheet which includes Basic integration, trigonometric, Inverse Trigonometric, Hyperbolic Integration, Inverse Hyperbolic, Exponential, Logarithmic Integration and Standard Integration formulas. You will also learn Methods of Integration.
Basic Integration Formulas
Some basic integral formulas are given below; these are derived from the Fundamental Theorem of Calculus. You should memorize these formulas. It is common to forget small details or concepts while solving problems; however, if you regularly study topics like integration formulas and solve related problems, you will easily learn how to solve calculus problems.
- \(\int k\,dx = kx + C\)
- \(\int x^n\,dx = \frac{x^{n+1}}{n+1}+C,\; n\neq -1\)
- \(\int \frac{1}{x}\,dx = \ln|x|+C\)
- \(\int e^x\,dx = e^x+C\)
- \(\int a^x\,dx = \frac{a^x}{\ln a}+C\)
- \(\int \sqrt{x}\,dx = \frac{2}{3}x^{3/2}+C\)
Trigonometric Integration
- \(\int \sin x\,dx = -\cos x+C\)
- \(\int \cos x\,dx = \sin x+C\)
- \(\int \tan x\,dx = \ln|\sec x|+C\)
- \(\int \cot x\,dx = \ln|\sin x|+C\)
- \(\int \sec x\,dx = \ln|\sec x+\tan x|+C\)
- \(\int \csc x\,dx = \ln|\csc x-\cot x|+C\)
- \(\int \sec^2x\,dx = \tan x+C\)
- \(\int \csc^2x\,dx = -\cot x+C\)
- \(\int \sec x\tan x\,dx = \sec x+C\)
- \(\int \csc x\cot x\,dx = -\csc x+C\)
Inverse Trigonometric Integration
- \(\int \frac{1}{\sqrt{1-x^2}}\,dx = \sin^{-1}x+C\)
- \(\int \frac{-1}{\sqrt{1-x^2}}\,dx = \cos^{-1}x+C\)
- \(\int \frac{1}{1+x^2}\,dx = \tan^{-1}x+C\)
- \(\int \frac{-1}{1+x^2}\,dx = \cot^{-1}x+C\)
- \(\int \frac{1}{|x|\sqrt{x^2-1}}\,dx = \sec^{-1}x+C\)
- \(\int \frac{-1}{|x|\sqrt{x^2-1}}\,dx = \csc^{-1}x+C\)
Hyperbolic Integration
- \(\int \sinh x\,dx = \cosh x+C\)
- \(\int \cosh x\,dx = \sinh x+C\)
- \(\int \tanh x\,dx = \ln(\cosh x)+C\)
- \(\int \coth x\,dx = \ln|\sinh x|+C\)
- \(\int \operatorname{sech}^2x\,dx = \tanh x+C\)
- \(\int \operatorname{csch}^2x\,dx = -\coth x+C\)
- \(\int \operatorname{sech}x\tanh x\,dx = -\operatorname{sech}x+C\)
- \(\int \operatorname{csch}x\coth x\,dx = -\operatorname{csch}x+C\)
Inverse Hyperbolic Integration
- \(\int \frac{1}{\sqrt{x^2+1}}\,dx = \sinh^{-1}x+C\)
- \(\int \frac{1}{\sqrt{x^2-1}}\,dx = \cosh^{-1}x+C\)
- \(\int \frac{1}{1-x^2}\,dx = \tanh^{-1}x+C\)
- \(\int \frac{1}{1-x^2}\,dx = \coth^{-1}x+C\)
- \(\int \frac{-1}{x\sqrt{1-x^2}}\,dx = \operatorname{sech}^{-1}x+C\)
- \(\int \frac{-1}{|x|\sqrt{1+x^2}}\,dx = \operatorname{csch}^{-1}x+C\)
Exponential and Logarithmic Integration
- \(\int e^{ax}\,dx=\frac{e^{ax}}{a}+C\)
- \(\int a^{x}\,dx=\frac{a^{x}}{\ln a}+C\)
- \(\int \ln x\,dx=x\ln x-x+C\)
- \(\int \log_a x\,dx=\frac{x\ln x-x}{\ln a}+C\)
Standard Integration
- \(\int \frac{1}{a^2+x^2}\,dx=\frac{1}{a}\tan^{-1}\frac{x}{a}+C\)
- \(\int \frac{1}{\sqrt{a^2-x^2}}\,dx=\sin^{-1}\frac{x}{a}+C\)
- \(\int \frac{1}{\sqrt{x^2+a^2}}\,dx=\ln|x+\sqrt{x^2+a^2}|+C\)
- \(\int \frac{1}{\sqrt{x^2-a^2}}\,dx=\ln|x+\sqrt{x^2-a^2}|+C\)
- \(\int \frac{1}{x^2-a^2}\,dx=\frac{1}{2a}\ln\left|\frac{x-a}{x+a}\right|+C\)
If you want to score well in AP Calculus AB/BC and Class 11/12 Maths, you will also need to learn and memorize derivative formulas.
Methods of Integration
- \(\int [f(x)\pm g(x)]dx=\int f(x)dx\pm\int g(x)dx\)
- \(\int kf(x)dx=k\int f(x)dx\)
- \(\int u\,dv=uv-\int v\,du\)
- Substitution: \(\int f(g(x))g'(x)\,dx=\int f(t)\,dt\)