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# Differentiation Integration Formula

Differentiation Integration Formula Sheet :

Differentiation Integration Formula Sheet is very important for learning students, it gives an quick referral over the subject during assessments or exam. Summarizing the formula in pdf is very simple and easy understanding Differentiation and integration ideas are used in combining chain rule and to carry out further integrals.

Differential Formula

A list of numerical derivative is given below for solving higher derivative function or problems.

1. $\frac{d}{dx}$ k = 0

2. $\frac{d}{dx}$ k.f(x) = kf (x)

3. $\frac{d}{dx}$ f(x)+g(x) = f'(x)+g'(x)

4. $\frac{d}{dx}$ f(g(x)) = f'(g(x)) .g'(x)

5. $\frac{d}{dx}x^{n}=nx^{n-1}$

6. $\frac{d}{dx}$ sin x = cos x

7. $\frac{d}{dx}$ cos x = -sin x

8. $\frac{d}{dx}$ tan x = sec2x

9. $\frac{d}{dx}$ cot x = -csc2x

10. $\frac{d}{dx}$ sec x = sec x tan x

11. $\frac{d}{dx}$ csc x = -csc x cot x

12. $\frac{d}{dx}$ ax – ax loga

13. $\frac{d}{dx}$ ex = ex

14. $\frac{d}{dx}$ log x = $\frac{1}{x}$

15. $\frac{d}{dx}$ sin-1x = $\frac{1}{\sqrt{1-x^2}}$

16. $\frac{d}{dx}$ cos-1x =  $\frac{-1}{\sqrt{1-x^2}}$

17. $\frac{d}{dx}$ tan-1x = $\frac{1}{x^2+1}$

18. $\frac{d}{dx}$ cot-1x = $\frac{-1}{x^2+1}$

19. $\frac{d}{dx}$ sec-1x = $\frac{1}{\left | x \right |\sqrt{x^2-1}}$

20. $\frac{d}{dx}$ csc-1x = $\frac{-1}{\left | x \right |\sqrt{x^2-1}}$

Integration Formula :

list of trigonometry integral formula are given below by using reduction method.

1. $\int$ dx = x + c

2. $\int$ a dx = ax + c

3. $\int$ (u + v) dx = $\int$ u dx + $\int$ vdx

4. $\int$ $x^n$ dx = $\frac{x^{n+1}}{n+1}$ + c

5. $\int$ $(ax+b)^n$ dx = $\frac{x^{ax+b}}{a(n+1)}$ + c

6. $\frac{\int 1}{x}$ dx = log x+c

7. $\int$ $e^x$ dx = $e^x$ + c

8. $\int$ $e^{mx}$ dx = $\frac{e^{mx}}{m}$ + c

9. $\int$ sin x dx = - cos x + c

10. $\int$ cos x dx = sin x + c

11. $\int$ tan x dx = $\frac{1}{sec x}$ + c

12. $\int$ cot x dx = $\frac{-1}{csc x}$ + c

13. $\int$ sec x dx = log (sec x + tan x) + c

14. $\int$ csc x dx = - log (csc x + cot x) + c

15. $\int$ $sec^2$ x dx = tan x + c

16. $\int$ $csc^2$ x dx = -cot x + c

17. $\int$ sec x tan x dx = secx + c

18. $\int$ csc x cot x dx = - csc x + c

19. $\int$ $sech^2$ x dx = tanh x + c

20. $\int$ $csch^2$ x dx = -coth x + c

21. $\int$ sech x tanh x dx = -sech x + c

22. $\int$ csch x coth x dx = -csch x + c

23. $\int$ sin mx dx = -$\frac{cosmx}{m}$ + c

24. $\int$ $sec^2$ mx dx = $\frac{tanmx}{m}$ + c

25. $\int$ $cosec^2$ mx dx = $\frac{-cotmx}{m}$ + c

26. $\int$ sec mx tan mx dx = $\frac{secmx}{m}$ + c

27. $\int$ cosec mx cot mx dx = $\frac{-cosecmx}{m}$ + c

Important Integrals:-

28. $\int$ $\frac{1}{\sqrt{1-x^2}}$ dx = sin-1 x + c

29. $\int$ $\frac{1}{\sqrt{a^2 - x^2}}$ dx = sin-1 $\frac{x}{a}$ + c (or) cos-1 $\frac{x}{a}$ + c

30. $\int$ $\frac{f'(x)}{\sqrt{a^2-f(x)^2}}$ dx = sin-1$\frac{f(x)}{a}$ + c

31. $\int$ $\frac{1}{\sqrt{x^2 + 1}}$ dx = tan-1 x + c   (or ) - cot-1 x + c

32. $\int$ $\frac{1}{\sqrt{x^2 + a^2}}$  dx = $\frac{1}{a}$ tan-1 $\frac{x}{a}$ + c (or)  - $\frac{1}{a}$ cot-1 $\frac{x}{a}$ + c

33. $\int$ $(\frac{1}{x\sqrt{x^2-1}})$ dx = sec-1 x + c (or ) – cosec-1 x + c

34. $\int$ $(\frac{1}{x\sqrt{x^2-a^2}})$ dx = $\frac{1}{a}$ sec-1 $\frac{x}{a}$ + c (or) - $\frac{1}{a}$ cosec-1 $\frac{x}{a}$ + c

35. $\int$ $\frac{1}{\sqrt{x^2+1}}$ dx = sinh-1 x + c  (or) loge $(x+\sqrt{(x^2+1)})$

36. $\int$ $\frac{1}{\sqrt{x^2+a^2}}$ dx = sinh-1 $\frac{x}{a}$ + c  (or) loge $( x+\sqrt{(x^2+a^2)})$

37. $\int$ $\frac{f'(x)}{\sqrt{f(x^2)-a^2}}$ dx = cosh-1  $\frac{f(x)}{a}$ + c

38. $\sqrt{\int a^2+x^2}$ dx = $\frac{x}{2}\sqrt{a^2+x^2}+\frac{a^2}{2}$ sinh-1 $\frac{x}{a}$ + c

39. $\sqrt{\int a^2-x^2}$ dx = $\frac{x}{2}\sqrt{a^2-x^2}$ + $\frac{a^2}{2}$ sinh-1 $\frac{x}{a}$ + c

40. $\sqrt{\int x^2-a^2}$ dx = $\frac{x}{2}\sqrt{x^2-a^2}$ - $\frac{a^2}{2}$ cosh-1 $\frac{x}{a}$ + c