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}

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