Formulas of Integration

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

2) $\int {adx = ax + c}$ where $a$ is 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) $\int {{a^x}dx = \frac{{{a^x}}}{{\ln a}} + c}$

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

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

41) $\int {\frac{1}{{x\sqrt {{a^2} – {x^2}} }}dx = – \frac{1}{a}\sec {{\text{h}}^{ – 1}}\frac{x}{a} + c}$ or $– \frac{1}{a}\ln \left( {\frac{{a + \sqrt {{a^2} – {x^2}} }}{x}} \right) + c$

42) $\int {\frac{1}{{x\sqrt {{x^2} – {a^2}} }}dx = \frac{1}{a}{{\sec }^{ – 1}}\frac{x}{a} + c}$

43) $\int {\frac{1}{{x\sqrt {{x^2} + {a^2}} }}dx = – \frac{1}{a}\csc {{\text{h}}^{ – 1}}\frac{x}{a} + c}$ or $- \frac{1}{a}\ln \left( {\frac{{a + \sqrt {{x^2} + {a^2}} }}{x}} \right) + c$

44) $\int {\sqrt {{a^2} – {x^2}} } dx = \frac{1}{2}x\sqrt {{a^2} – {x^2}} + \frac{{{a^2}}}{2}{\sin ^{ – 1}}\frac{x}{a} + c$

45) $\int {\sqrt {{x^2} – {a^2}} dx = \frac{1}{2}x\sqrt {{x^2} – {a^2}} – \frac{{{a^2}}}{2}{{\cosh }^{ – 1}}\frac{x}{a} + c}$ or $\frac{1}{2}x\sqrt {{x^2} – {a^2}} – \frac{{{a^2}}}{2}\ln \left( {x + \sqrt {{x^2} – {a^2}} } \right) + c$

46) $\int {\sqrt {{x^2} + {a^2}} dx = \frac{1}{2}x\sqrt {{x^2} + {a^2}} + \frac{{{a^2}}}{2}{{\sinh }^{ – 1}}\frac{x}{a} + c}$ or $\frac{1}{2}x\sqrt {{x^2} + {a^2}} + \frac{{{a^2}}}{2}\ln \left( {x + \sqrt {{x^2} + {a^2}} } \right) + c$

47) $\int {{e^{ax}}\sin (bx + c)dx = \frac{{{e^{ax}}}}{{{a^2} + {b^2}}}\left[ {a\sin (bx + c) – b\cos (bx + c)} \right]}$

48) $\int {{e^{ax}}\cos (bx + c)dx = \frac{{{e^{ax}}}}{{{a^2} + {b^2}}}\left[ {a\cos (bx + c) + b\sin (bx + c)} \right]}$

49) $\int {\sin mx\cos nxdx = – \frac{{\cos (m + n)x}}{{2(m + n)}}} – \frac{{\cos (m – n)x}}{{2(m – n)}} + c$

50) $\int {\sin mx\sin nxdx = – \frac{{\sin (m + n)x}}{{2(m + n)}}} + \frac{{\sin (m – n)x}}{{2(m – n)}} + c$

51) $\int {\cos mx\cos nxdx = \frac{{\sin (m + n)x}}{{2(m + n)}}} + \frac{{\sin (m – n)x}}{{2(m – n)}} + c$

52) $\int {{{\sin }^{ – 1}}xdx = x{{\sin }^{ – 1}}x + \sqrt {1 – {x^2}} + c}$

53) $\int {{{\cos }^{ – 1}}xdx = x{{\cos }^{ – 1}}x – \sqrt {1 – {x^2}} + c}$

54) $\int {{{\tan }^{ – 1}}xdx = x{{\tan }^{ – 1}}x – \frac{1}{2}\ln (1 + {x^2}) + c}$

55) $\int {{{\cot }^{ – 1}}xdx = x{{\cot }^{ – 1}}x + \frac{1}{2}\ln (1 + {x^2}) + c}$

56) $\int {{{\sec }^{ – 1}}xdx = x{{\sec }^{ – 1}}x – \ln \left( {x + \sqrt {{x^2} – 1} } \right) + c}$

57) $\int {{{\csc }^{ – 1}}xdx = x{{\csc }^{ – 1}}x + \ln \left( {x + \sqrt {{x^2} – 1} } \right) + c}$

58) $\int {\frac{1}{{a + b\sin x}}dx = \frac{2}{{\sqrt {{a^2} – {b^2}} }}{{\tan }^{ – 1}}\left( {\frac{{a{{\tan }^{ – 1}}\frac{x}{2} + b}}{{\sqrt {{a^2} – {b^2}} }}} \right) + c}$ if ${a^2} > {b^2}$

59) $\int {\frac{1}{{a + b\sin x}}dx = \frac{1}{{\sqrt {{a^2} – {b^2}} }}\ln \left( {\frac{{a\tan \frac{x}{a} + b – \sqrt {{b^2} – {a^2}} }}{{a\tan \frac{x}{a} + b + \sqrt {{b^2} – {a^2}} }}} \right) + c}$ if ${a^2} < {b^2}$

60) $\int {\frac{1}{{a + b\cos x}}dx = \frac{2}{{\sqrt {{a^2} – {b^2}} }}{{\tan }^{ – 1}}\left( {\sqrt {\frac{{a – b}}{{a + b}}} \tan \frac{x}{2}} \right) + c}$ if ${a^2} > {b^2}$

61) $\int {\frac{1}{{a + b\cos x}}dx = \frac{1}{{\sqrt {{a^2} – {b^2}} }}\ln \left( {\frac{{\sqrt {b + a} + \tan \frac{x}{2}\sqrt {b – a} }}{{\sqrt {b + a} – \tan \frac{x}{2}\sqrt {b – a} }}} \right) + c}$ if ${a^2} < {b^2}$

62) $\int {\frac{1}{{a + b\sinh x}}dx = \frac{1}{{\sqrt {{a^2} + {b^2}} }}\ln \left( {\frac{{\sqrt {{a^2} + {b^2}} + a\tanh \frac{x}{2} – b}}{{\sqrt {{a^2} + {b^2}} – a\tanh \frac{x}{2} + b}}} \right) + c}$

63) $\int {\frac{1}{{a + b\cosh x}}dx = \frac{{\sqrt {a + b} + \sqrt {a – b} \tanh \frac{x}{2}}}{{\sqrt {a + b}- \sqrt {a – b} \tanh \frac{x}{2}}} + c}$ if $a > b$

64) $\int {\frac{1}{{a + b\cosh x}}dx = \frac{2}{{\sqrt {{b^2} – {a^2}} }}{{\tan }^{ – 1}}\sqrt {\frac{{b – a}}{{b + a}}} {{\tanh }^{ – 1}}\frac{x}{2} + c}$ if $a < b$