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 $$a < b$$