Basic Integral Formulas

1) \int {1dx = x + c}

2) \int {adx = ax + c} Where ais any constant.

3) \int {{x^n}dx = \frac{{{x^{n + 1}}}}{{n + 1}} + c}

4) \int {{{[f(x)]}^n}f'(x)dx = \frac{{{{[f(x)]}^{n + 1}}}}{{n + 1}}} + c

5) \int {\frac{1}{x}dx = \ln x + c}

6) \int {\frac{{f'(x)}}{{f(x)}}dx = \ln f(x) + c}

7) \int {{a^x}dx = \frac{{{a^x}}}{{\ln x}} + c}

8) {\int a ^{f(x)}}dx = \frac{{{a^{f(x)}}}}{{\ln a}} + c

9) \int {{e^x}dx = {e^x} + c}

10) \int {{e^{f(x)}}dx = {e^{f(x)}} + c}

11) \int {af(x)dx = a\int {f(x)} }

12) \int {[f(x) \pm g(x)]dx = \int {f(x)dx \pm \int {g(x)dx} } }

13) \int {f(x) \cdot g(x)dx = f(x)\left( {\int {g(x)dx} } \right) - \left[ {f'(x)\left( {\int {g(x)dx} } \right)} \right]dx}

14) \int {\ln xdx = x(\ln x - 1) + c}

15) \int {\sin xdx = - \cos x + c}

16) \int {\cos xdx = \sin x + c}

17) \int {\tan xdx = \ln \sec x} + c or  - \ln \cos x + c

18) \int {\cot xdx = \ln \sin x + c}

19) \int {\sec xdx = \ln (\sec x + \tan x) + c} or \ln \tan \left( {\frac{x}{2} + \frac{\pi }{4}} \right) + c

20) \int {\csc xdx = \ln (\csc x - \cot x) + c} or \ln \tan \frac{x}{2} + c

21) \int {{{\sec }^2}xdx = \tan x + c}

22) \int {{{\csc }^2}xdx = - \cot x + c}

23) \int {\sec x\tan xdx = \sec x + c}

24) \int {\csc x\cot xdx = - \csc x + c}

25) \int {\sinh xdx = \cosh x + c}

26) \int {\cosh xdx = \sinh x + c}

27) \int {\tanh xdx = \ln \cosh x + c}

28) \int {\coth xdx = \ln \sinh x + c}

29) \int {\sec {\text{h}}xdx = {{\tan }^{ - 1}}(\sinh x) + c}

30) \int {\csc {\text{h}}xdx = - {{\coth }^{ - 1}}(\cosh x)}

31) \int {\sec {{\text{h}}^2}xdx = \tanh x + c}

32) \int {\csc {{\text{h}}^2}xdx = - \coth x + c}

33) \int {\sec {\text{h}}x\tanh xdx = - \sec {\text{h}}x + c}

34) \int {\csc {\text{h}}x\coth xdx = - \csc {\text{h}}x + c}

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

36) \int {\frac{1}{{\sqrt {{x^2} - {a^2}} }}dx = {{\cosh }^{ - 1}}\frac{x}{a}} + c or \ln (x + \sqrt {{x^2} - {a^2}} ) + c

37) \int {\frac{1}{{\sqrt {{x^2} + {a^2}} }}dx = {{\sinh }^{ - 1}}\frac{x}{a} + c} or \ln (x + \sqrt {{x^2} + {a^2}} ) + c

38) \int {\frac{1}{{{a^2} - {x^2}}}dx = \frac{1}{a}{{\tanh }^{ - 1}}\frac{x}{a} + c} or \frac{1}{{2a}}\ln \left( {\frac{{a + x}}{{a - x}}} \right) + c

39) \int {\frac{1}{{{x^2} - {a^2}}}dx = - \frac{1}{a}{{\coth }^{ - 1}}\frac{x}{a} + c} or \frac{1}{{2a}}\ln \left( {\frac{{x - a}}{{x + a}}} \right) + c

40) \int {\frac{1}{{{x^2} + {a^2}}}dx = \frac{1}{a}{{\tan }^{ - 1}}\frac{x}{a} + c}