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Genel İntegraller

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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