Integration formula list
Integration is on of the most important topic for higher class mathematics and physics. It is a fundamental, essential operation of calculus, and serves as a tool to solve problems in mathematics and physics involving the area of an arbitrary shape, the length of a curve, moments and center of mass, exponential growth and decay , probability,and the volume of a solid, among others.
Here I am going to share a list about Integration Formula. Readers can refer these to solve their problems.
Topics included:
- Some basic formula for Integration
- Some special formula for Integration
- Integration by parts
- Formula for definite Integration
- PDF formula sheet
Some Basic Formula of Integration
- \(\int x^{n} dx =\frac{x^{n+1}}{n+1}+C\)
- \(\int cosx dx=sinx + C\)
- \(\int sinx dx=-sinx +C\)
- \(\int sec^2x=tanx+C\)
- \(\int cosec^2x dx=-cotx +C\)
- \(\int secx tanx dx=secx+C\)
- \(\int cosecx cotx dx=-cosecx+ C\)
Some Special Formula of Integration
- \(\int \frac{dx}{\sqrt{1-x^2}}=sin^{-1}x +C \)
- \(\int\frac{dx}{\sqrt{1-x^2}}=-cos^{-1}x+C\)
- \(\int\frac{dx}{1+x^2}=tan^{-1}x +C\)
- \(\int\frac{dx}{1+x^2}=-cot^{-1}x +C\)
- \(\int e^x dx=e^x+C\)
- \(\int a^xdx=\frac{a^x}{loga}+C\)
- \(\int\frac{dx}{x\sqrt{x^2-1}}=sec^{-1}x+C\)
- \(\int\frac{dx}{x\sqrt{x^2-1}}=-cosec^{-1}x+C\)
- \(\int\frac{1}{x}dx=log|x|+C\)
- \(\int tanx dx=log |secx|+C\)
- \(\int cotx dx=log|sinx|+ C\)
- \(\int secx dx=log|secx+tanx|+ C\)
- \(\int cosecx dx=log|cosecx-cotx|+C\)
- \(\int\frac{dx}{x^2-a^2}=\frac{1}{2a}log|\frac{x-a}{x+a}|+C\)
- \(\int\frac{dx}{a^2-x^2}=\frac{1}{2a}log|\frac{a+x}{a-x}|+C\)
- \(\int \frac{dx}{x^2+a^2}=\frac{1}{a}tan^{-1}\frac{x}{a}+C\)
- \(\int\frac{dx}{\sqrt{x^2-a^2}}=log|x+\sqrt{x^2-a^2}|+C\)
- \(\int\frac{dx}{\sqrt{a^2-x^2}}=sin^{-1}\frac{x}{a}+C\)
- \(\int\frac{dx}{\sqrt{x^2+a^2}}=log|x+\sqrt{x^2+a^2}|+C\)
- \(\int \sqrt(x^2-a^2)dx=\frac{x}{2}\sqrt(x^2-a^2)-\frac{a^2}{2}log|x+\sqrt{x^2-a^2}+C\)
- \(\int\sqrt{x^2+a^2} dx =\frac{x}{2}\sqrt{x^2+a^2}+\frac{a^2}{2}log|x+\sqrt{x^2+a^2} +C\)
- \(\int\sqrt{a^2-x^2} dx=\frac{x}{2}\sqrt{a^2-x^2}+\frac{a^2}{2}sin^{-1}\frac{x}{a}+C\)
Integration by Parts
- \(\int f(x) g(x) dx=f(x)\int g(x)dx-\int{f(x)\int g(x) dx}dx\)
- \(\int e^x[f(x)+ f’(x)]dx=\int e^x f(x) dx +C\)
Formula for Definite Integration
- If \(\int f(x)dx=G(x)\) so,\(\int_{a}^{b}f(x)dx=G(b)-G(a)\)
- \(\int_{a}^{b} f(x) dx=\int_{a}^{b}f(t)dt\)
- \(\int_{a}^{b}f(x)dx=-\int_{b}^{a}f(x)dx\)
- \(\int_{a}^{b}f(x)dx=\int_{a}^{c}f(x)dx+\int_{c}{b}f(x)dx\)
- \(\int_{a}^{b}f(x)dx=\int_{a}^{b}f(a+b-x)dx\)
- \(\int_{0}^{a}f(x)dx=\int_{0}^{a}f(a-x)dx\)
- \(\int_{0}^{2a}f(x)dx=\int_{0}^{a} f(x)dx+\int_{0}^{a}f(2a-x)dx\)
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