Integration Formula

Integration

Integration is the operation of calculating the area between the curve of a function and the x-axis. Integral also includes antiderivative and primitive. Integration works by transforming a function into another function respectively.

Some of the important integration formulas are listed below:-

$1. \displaystyle{\int x^n dx = \dfrac{x^{n + 1}}{n + 1}} \\[3mm] \displaystyle{\int (ax + b)^n = \dfrac{1}{a} \dfrac{(ax + b)^{n + 1}}{n + 1}, n \neq -1} \\[3mm] 2.\displaystyle{\int \dfrac{1}{x} dx = \log x} \\[3mm] \displaystyle{\int \dfrac{1}{ax + b} dx = \dfrac{1}{a} \log (ax + b)} \\[3mm] 3. \displaystyle{\int e^x dx = e^x} \\[3mm] \displaystyle{\int e^{ax + b} dx = \dfrac{1}{a}e^{ax + b}} \\[3mm] 4. \displaystyle{\int a^x dx = \dfrac{a^x}{\log_e a}} \\[3mm] \displaystyle{\int a^{px + q} dx = \dfrac{1}{p} \dfrac{a^{px + q}}{\log_w a}}$ $5.\displaystyle{\int \sin x dx = -\cos x} \\[3mm] \displaystyle{\int \sin (ax + b) dx = -\dfrac{1}{a} \cos (ax + b)} \\[3mm] 6. \displaystyle{\int \cos dx = \sin x} \\[3mm] \displaystyle{\int \cos (ax + b) dx = \dfrac{1}{a} \sin (ax + b)} \\[3mm] 7.\displaystyle{\int sec^2 x dx = \tan x} \\[3mm] \displaystyle{\int \sec^2 (ax + b) dx = \dfrac{1}{a} \tan (ax + b)} \\[3mm] 8. \displaystyle{\int \csc^2 x dx = \tan x} \\ \displaystyle{\int \csc^2 (ax + b) dx =\dfrac{1}{a} \cot (ax + b)} \\[3mm] 9. \displaystyle{\int \sec x \tan x dx = \sec x} \\[3mm] \displaystyle{\int \sec (ax + b) \tan \left((ax + b) dx = \dfrac{1}{a} \sec (ax + b)\right)} \\[3mm] 10. \displaystyle{\int dx = x}$ $11. \displaystyle{\int \sin hx dx = \cos hx} \\[3mm] \displaystyle{\int \sin h(ax + b) dx = \dfrac{1}{a} \cos h (ax + b)} \\[3mm] 12. \displaystyle{\int \cos hx dx = \sin hx} \\[3mm] \displaystyle{\int \cos h(ax + b) dx = \dfrac{1}{a}\sin h(ax + b)} \\[3mm] 13. \displaystyle{\int sec h^2(ax + b) dx = \dfrac{1}{a}\tan h(ax + b)} \\[3mm] 14. \displaystyle{\int \csc h^2x dx = -\cot hx} \\[3mm] \displaystyle{\int \csc h^2(ax + b) dx = -\dfrac{1}{a} \cot h(ax + b)} \\[3mm] 15. \displaystyle{\int sec hx \tan hx dx = \sec hx} \\[3mm] \displaystyle{\int \sec h(ax + b) \tan h\left((ax + b) dx = \dfrac{1}{a} \sec h(ax + b) \right)}$ $16. \displaystyle{\int \csc hx \cot hx dx = -\csc hx} \\[3mm] \displaystyle{\int csc h(ax + b) \cot h(ax + b) dx = - \dfrac{1}{a} \csc h(ax + b)} \\[3mm] 17. \displaystyle{\int csc x dx = \log \tan \frac{x}{2} = \log (\csc x- \cot x)} \\[3mm] 18. \displaystyle{\int \sec x dx = \log \tan \left(\dfrac{\pi}{4} + \dfrac{x}{2} \right) = \log (\sec x + \tan x)} \\[3mm] 19. \displaystyle{\int \tan x dx = \log \sec x} \\[3mm] 20. \displaystyle{\int \cot x dx = \log \sin x}$ $21. \displaystyle{\int \tan hx dx = \log \cos hx} \\[3mm] 22. \displaystyle{\int \cot hx dx = \log \sin hx} \\[3mm] 23. \displaystyle{\int \dfrac{dx}{\sqrt{1- x^2}} = \sin ^{-1} x} \\ \displaystyle{\int \dfrac{dx}{\sqrt{a^2- x^2}} = \sin^{-1} x/a} \\[3mm] 24. \displaystyle{\int \dfrac{dx}{1 + x^2} = \tan^{-1}} \\[3mm] \displaystyle{\int \dfrac{dx}{a^2 + x^2} = \dfrac{1}{a} \tan^{-1}x/a} \\[3mm] 25. \displaystyle{\int \dfrac{dx}{x\sqrt{x^2 - 1}} = \sec^{-1} x } \\[3mm] \displaystyle{\int \dfrac{dx}{x\sqrt{x^2 - a^2}} = \dfrac{1}{a} \sec^{-1} x/a} \\[3mm] 26. \displaystyle{\int \dfrac{dx}{x^2 - a^2} = \dfrac{1}{2a} \log \dfrac{x - a}{x + a}(x > a)}$ $27. \displaystyle{\int \dfrac{dx}{a^2- x^2} = \dfrac{1}{2a} \log \dfrac{a + x}{a- x} (x < a)} \\ 28. \displaystyle{\int \dfrac{dx}{\sqrt{x^2 + a^2}} = \log (x + \sqrt{x^2 + a^2}) or, \cos h^{-1}x/a} \\[3mm] 29. \displaystyle{\int \dfrac{dx}{\sqrt{x^2- a^2}} = \log (x + \sqrt{a^2- x^2}) or, \cos h^{-1} x/a} \\[3mm] 30. \displaystyle{\int \sqrt{x^ + a^2}dx = \dfrac{x\sqrt{x^2 + a^2}}{2} + \dfrac{a^2}{2} \log (x + \sqrt{x^2 + a^2})} \\ 31. \displaystyle{\int \sqrt{x^2- a^2} dx = \dfrac{x\sqrt{x^2- a^2}}{2}- \dfrac{a^2}{2} \log (x + \sqrt{x^2- a^2})}$ $32. \displaystyle{\int \sqrt{a^2- x^2}dx = \dfrac{x \sqrt{a^2- x^2}}{2} + \dfrac{a^2}{2} \sin h^{-1}x/a} \\[3mm] 33. \displaystyle{\int u.v dx = u \int v dx- \int [(v dx) \times \dfrac{du}{dx}]dx} \\[3mm] 34. \displaystyle{\int e^{ax} \sin bx dx = e^{ax}\left(\dfrac{a \sin bx- b \cos bx}{a^2 + b^2} \right)} \\[3mm] 35. \displaystyle{\int e^{ax} \cos bx dx = e^{ax} \left(\dfrac{a \cos bx + b \sin bx}{a^2 + b^2} \right)} \\[3mm] 36. \displaystyle{\int \dfrac{f(x)}{f(x)} dx = \log f(x)}$