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Hesap Modülleri

### Genel Integraller

1.
$$\int{adx}=ax$$

2.
$$\int{af(x)dx}=a\int{f(x)dx}$$

3.
$$\int{(u\mp v\mp w)dx}=\int{udx}\mp\int{vdx}\mp\int{wdx}$$

4.
$$\int{udv}=uv-\int{vdu}$$

5.
$$\int{f(ax)dx}=\frac{1}{a}\int{f(u)du}$$

6.
$$\int\{F{f(x)\}dx}=\int{F(u)\frac{dx}{du}du}=\int{\frac{F(u)}{f'(x)}du}$$

7.
$$\int{u^ndu}=\frac{u^{n+1}}{n+1}\quad;\quad n\neq-1$$

8.
$$\int{\frac{du}{u}}=\ln|u|$$

9.
$$\int{e^{u}du}=e^{u}$$

10.
$$\int{a^{u}du}=\int{e^{u\ln a}du}=\frac{e^{u\ln a}}{\ln a}=\frac{a^u}{\ln a}\quad;\quad a>0 \quad;\quad a \neq 1$$

11.
$$\int{\sin udu}=-\cos u$$

12.
$$\int{\cos udu}=\sin u$$

13.
$$\int{\tan udu}=\ln\sec u=-\ln\cos u$$

14.
$$\int{\cot udu}=\ln\sin u$$

15.
$$\int{\sec udu}=\ln(\sec u+\tan u)=\ln\tan\left(\frac{u}{2}+\frac{\pi}{4}\right)$$

16.
$$\int{\csc udu}=\ln(\csc u-\cot u)=\ln\tan\frac{u}{2}$$

17.
$$\int{\sec^2 udu}=\tan u$$

18.
$$\int{\csc^2 udu}=-\cot u$$

19.
$$\int{\tan^2 udu}=\tan{u}-u$$

20.
$$\int{\cot^2 udu}=-\cot{u}-u \quad;\quad$$

21.
$$\int{\sin^2 udu}=\frac{u}{2}-\frac{\sin 2u}{4}=\frac{1}{2}(u-\sin u\cos u)$$

22.
$$\int{\cos^2 udu}=\frac{u}{2}+\frac{\sin 2u}{4}=\frac{1}{2}(u+\sin u\cos u)$$

23.
$$\int{\sec u\tan udu}=\sec u$$

24.
$$\int{\csc u\cot udu}=-\csc u$$

25.
$$\int{\sinh udu}=\cosh u$$

26.
$$\int{\cosh udu}=\sinh u$$

27.
$$\int{\tanh udu}=\ln\cosh u$$

28.
$$\int{\coth udu}=\ln\sinh u$$

29.
$$\int{\text{sech} udu}=\sin^{-1}(\tanh u) \;\;\text{ veya}\;\; 2\tan^{-1}e^u$$

30.
$$\int{\text{csch} udu}=\ln \tanh\frac{u}{2} \;\; \text{veya}\;\; -\coth^{-1}e^u$$

31.
$$\int{\text{sech}^2 udu}=\tanh u$$

32.
$$\int{\text{csch}^2 udu}=-\coth u$$

33.
$$\int{\tanh^2 udu}=u-\tanh u$$

34.
$$\int{\coth^2 udu}=u-\coth u$$

35.
$$\int{\sinh^2 udu}=\frac{\sinh 2u}{4}-\frac{u}{2}=\frac{1}{2}(\sinh u\cosh u-u)$$

36.
$$\int{\cosh^2 udu}=\frac{\sinh 2u}{4}+\frac{u}{2}=\frac{1}{2}(\sinh u\cosh u+u)$$

37.
$$\int{\text{sech} u\tanh udu}=-sech u$$

38.
$$\int{\text{csch} u\coth udu}=-csch u$$

39.
$$\int{\frac{du}{u^{2}+a^{2}}}=\frac{1}{a}\tan^{-1}\frac{u}{a}$$

40.
$$\int{\frac{du}{u^{2}-a^{2}}}=\frac{1}{2a}\ln\left(\frac{u-a}{u+a}\right)=-\frac{1}{a}\coth^{-1}\frac{u}{a}\quad;\quad u^{2}>a^{2}$$

41.
$$\int{\frac{du}{a^{2}-u^{2}}}=\frac{1}{2a}\ln\left(\frac{a+u}{a-u}\right)=\frac{1}{a}\tanh^{-1}\frac{u}{a}\quad;\quad u^{2}< a^{2}$$

42.
$$\int{\frac{du}{\sqrt{a^{2}-u^{2}}}}=\sin^{-1}\frac{u}{a}$$

43.
$$\int \frac{du}{\sqrt{u^{2}+a^{2}}} = \ln \left( u+ \sqrt{u^{2} + a^{2}} \right)$$

44.
$$\int \frac{du}{\sqrt{u^{2}-a^{2}}} = \ln \left( u + \sqrt{u^{2} - a^{2}} \right)$$

45.
$$\int{\frac{du}{u\sqrt{u^{2}-a^{2}}}}=\frac{1}{a}sec^{-1}\left|\frac{u}{a}\right|$$

46.
$$\int{\frac{du}{u\sqrt{u^{2}+a^{2}}}}=-\frac{1}{a}\ln\left(\frac{a+\sqrt{u^{2}+a^{2}}}{u}\right)$$

47.
$$\int{\frac{du}{u\sqrt{a^{2}-u^{2}}}}=-\frac{1}{a}\ln\left(\frac{a+\sqrt{u^{2}-a^{2}}}{u}\right)$$

48.
$$\int{f^{(n)}gdx}=f^{(n-1)}g-f^{(n-2)}g'+f^{(n-3)}g''-\cdots(-1)^{n}\int{fg^{(n)}dx}$$

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