Integration Formula List

Integration formula list

Integration Formula

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

  1. \(\int x^{n} dx =\frac{x^{n+1}}{n+1}+C\)
  2. \(\int cosx dx=sinx + C\)
  3. \(\int sinx dx=-sinx +C\)
  4. \(\int sec^2x=tanx+C\)
  5. \(\int cosec^2x dx=-cotx +C\)
  6. \(\int secx tanx dx=secx+C\)
  7. \(\int cosecx cotx dx=-cosecx+ C\)

Some Special Formula of Integration

  1. \(\int \frac{dx}{\sqrt{1-x^2}}=sin^{-1}x +C \)
  2. \(\int\frac{dx}{\sqrt{1-x^2}}=-cos^{-1}x+C\)
  3. \(\int\frac{dx}{1+x^2}=tan^{-1}x +C\)
  4. \(\int\frac{dx}{1+x^2}=-cot^{-1}x +C\)
  5. \(\int e^x dx=e^x+C\)
  6. \(\int a^xdx=\frac{a^x}{loga}+C\)
  7. \(\int\frac{dx}{x\sqrt{x^2-1}}=sec^{-1}x+C\)
  8. \(\int\frac{dx}{x\sqrt{x^2-1}}=-cosec^{-1}x+C\)
  9. \(\int\frac{1}{x}dx=log|x|+C\)
  10. \(\int tanx dx=log |secx|+C\)
  11. \(\int cotx dx=log|sinx|+ C\)
  12. \(\int secx dx=log|secx+tanx|+ C\)
  13. \(\int cosecx dx=log|cosecx-cotx|+C\)
  14. \(\int\frac{dx}{x^2-a^2}=\frac{1}{2a}log|\frac{x-a}{x+a}|+C\)
  15. \(\int\frac{dx}{a^2-x^2}=\frac{1}{2a}log|\frac{a+x}{a-x}|+C\)
  16. \(\int \frac{dx}{x^2+a^2}=\frac{1}{a}tan^{-1}\frac{x}{a}+C\)
  17. \(\int\frac{dx}{\sqrt{x^2-a^2}}=log|x+\sqrt{x^2-a^2}|+C\)
  18. \(\int\frac{dx}{\sqrt{a^2-x^2}}=sin^{-1}\frac{x}{a}+C\)
  19. \(\int\frac{dx}{\sqrt{x^2+a^2}}=log|x+\sqrt{x^2+a^2}|+C\)
  20. \(\int \sqrt(x^2-a^2)dx=\frac{x}{2}\sqrt(x^2-a^2)-\frac{a^2}{2}log|x+\sqrt{x^2-a^2}+C\)
  21. \(\int\sqrt{x^2+a^2} dx =\frac{x}{2}\sqrt{x^2+a^2}+\frac{a^2}{2}log|x+\sqrt{x^2+a^2} +C\)
  1. \(\int\sqrt{a^2-x^2} dx=\frac{x}{2}\sqrt{a^2-x^2}+\frac{a^2}{2}sin^{-1}\frac{x}{a}+C\)

Integration by Parts

  1. \(\int  f(x) g(x) dx=f(x)\int g(x)dx-\int{f(x)\int g(x) dx}dx\)
  2. \(\int e^x[f(x)+ f’(x)]dx=\int e^x f(x) dx +C\)

Formula for Definite Integration

  1. If \(\int f(x)dx=G(x)\) so,\(\int_{a}^{b}f(x)dx=G(b)-G(a)\)
  2. \(\int_{a}^{b} f(x) dx=\int_{a}^{b}f(t)dt\)
  3. \(\int_{a}^{b}f(x)dx=-\int_{b}^{a}f(x)dx\)
  4. \(\int_{a}^{b}f(x)dx=\int_{a}^{c}f(x)dx+\int_{c}{b}f(x)dx\)
  5. \(\int_{a}^{b}f(x)dx=\int_{a}^{b}f(a+b-x)dx\)
  6. \(\int_{0}^{a}f(x)dx=\int_{0}^{a}f(a-x)dx\)
  7. \(\int_{0}^{2a}f(x)dx=\int_{0}^{a} f(x)dx+\int_{0}^{a}f(2a-x)dx\)
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