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 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}

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

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