İntegral Alma Kuralları (Formülleri)

Arkadaşlar sizlere İntegral alma formülleri (kuralları) ile ilgili bir ders notu hazırladım umarım yardımcı olur.


İntegral Alma Kuralları (Formülleri)

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

2) \int {adx = ax + c} (Sabit sayı olduğunda.)

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} (üstel fonkiyonun integrali)

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} (lnx integrali)

15) \int {\sin xdx = – \cos x + c}

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

17) \int {\tan xdx = \ln \sec x} + c veya – \ln \cos x + c

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

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

20) \int {\csc xdx = \ln (\csc x – \cot x) + c} veya \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 veya {\cos ^{ – 1}}\frac{x}{a} + c

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

37) \int {\frac{1}{{\sqrt {{x^2} + {a^2}} }}dx = {{\sinh }^{ – 1}}\frac{x}{a} + c} veya \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} veya \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} veya \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}

veya

\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}

veya

\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}

veya

\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}

veya

\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}

( {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}

{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}

({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}

({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}
(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} ,(a < b)

Sizlere Musait zamanlarında bu şekilde notlar hazırlayacağım.Merak ettiklerinizi sorabilirsiniz Tüm arkadaşlara başarılar dilerim.

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Anladım teşekkürler…

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