 Matematik
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Genel İntegraller
- 1.
- \(\small \displaystyle\int{adx}=ax\)
- 2.
- \(\small \displaystyle\int{a\displaystyle f(x)dx}=a\displaystyle\int{f(x)dx}\)
- 3.
- \(\small \displaystyle\int{(u\mp v\mp w)dx}=\displaystyle\int{udx}\mp\displaystyle\int{vdx}\mp\displaystyle\int{wdx}\)
- 4.
- \(\small \displaystyle\int{udv}=uv-\displaystyle\int{vdu}\)
- 5.
- \(\small \displaystyle\int{f(ax)dx}=\frac{1}{a}\displaystyle\int{f(u)du}\)
- 6.
- \(\small \displaystyle\int\{F{f(x)\}dx}=\displaystyle\int{F(u)\displaystyle\frac{dx}{du}du}=\displaystyle\int{\displaystyle\frac{F(u)}{f'(x)}du}\)
- 7.
- \(\small \displaystyle\int{u^ndu}=\frac{u^{n+1}}{n+1}\quad;\quad n\neq-1\)
- 8.
- \(\small \displaystyle\int{\displaystyle\frac{du}{u}}=\displaystyle\ln|u|\)
- 9.
- \(\small \displaystyle\int{e^{u}du}=e^{u}\)
- 10.
- \(\small \displaystyle\int{a^{u}du}=\displaystyle\int{e^{u\ln a}du}=\displaystyle\frac{e^{u\ln a}}{\displaystyle\ln a}=\displaystyle\frac{a^u}{\ln a}\quad;\quad a>0 \quad;\quad a \neq 1\)
- 11.
- \(\small \displaystyle\int{\displaystyle\sin udu}=-\displaystyle\cos u\)
- 12.
- \(\small \displaystyle\int{\displaystyle\cos udu}=\displaystyle\sin u\)
- 13.
- \(\small \displaystyle\int{\displaystyle\tan udu}=\displaystyle\ln\displaystyle\sec u=-\displaystyle\ln\displaystyle\cos u\)
- 14.
- \(\small \displaystyle\int{\displaystyle\cot udu}=\displaystyle\ln\displaystyle\sin u\)
- 15.
- \(\small \displaystyle\int{\displaystyle\sec udu}=\displaystyle\ln(\displaystyle\sec u+\tan u)=\displaystyle\ln\displaystyle\tan\left(\frac{u}{2}+\frac{\pi}{4}\right)\)
- 16.
- \(\small \displaystyle\int{\displaystyle\csc udu}=\displaystyle\ln(\displaystyle\csc u-\displaystyle\cot u)=\displaystyle\ln\displaystyle\tan\displaystyle\frac{u}{2}\)
- 17.
- \(\small \displaystyle\int{\displaystyle\sec^2 udu}=\displaystyle\tan u\)
- 18.
- \(\small \displaystyle\int{\displaystyle\csc^2 udu}=-\displaystyle\cot u\)
- 19.
- \(\small \displaystyle\int{\displaystyle\tan^2 udu}=\displaystyle\tan{u}-u\)
- 20.
- \(\small \displaystyle\int{\displaystyle\cot^2 udu}=-\displaystyle\cot{u}-u \quad;\quad\)
- 21.
- \(\small \displaystyle\int{\displaystyle\sin^2 udu}=\displaystyle\frac{u}{2}-\displaystyle\frac{\displaystyle\sin 2u}{4}=\displaystyle\frac{1}{2}(u-\displaystyle\sin u\displaystyle\cos u)\)
- 22.
- \(\small \displaystyle\int{\displaystyle\cos^2 udu}=\displaystyle\frac{u}{2}+\displaystyle\frac{\displaystyle\sin 2u}{4}=\displaystyle\frac{1}{2}(u+\displaystyle\sin u\displaystyle\cos u)\)
- 23.
- \(\small \displaystyle\int{\displaystyle\sec u\displaystyle\tan udu}=\displaystyle\sec u\)
- 24.
- \(\small \displaystyle\int{\displaystyle\csc u\displaystyle\cot udu}=-\displaystyle\csc u\)
- 25.
- \(\small \displaystyle\int{\displaystyle\sinh udu}=\displaystyle\cosh u\)
- 26.
- \(\small \displaystyle\int{\displaystyle\cosh udu}=\displaystyle\sinh u\)
- 27.
- \(\small \displaystyle\int{\displaystyle\tanh udu}=\displaystyle\ln\displaystyle\cosh u\)
- 28.
- \(\small \displaystyle\int{\displaystyle\coth udu}=\displaystyle\ln\displaystyle\sinh u\)
- 29.
- \(\small \displaystyle\int{\displaystyle\text{sech} udu}=\displaystyle\sin^{-1}(\displaystyle\tanh u) \;\;\text{ veya}\;\; 2\displaystyle\tan^{-1}e^u\)
- 30.
- \(\small \displaystyle\int{\displaystyle\text{csch} udu}=\displaystyle\ln \displaystyle\tanh\displaystyle\frac{u}{2} \;\; \text{veya}\;\; -\displaystyle\coth^{-1}e^u\)
- 31.
- \(\small \displaystyle\int{\displaystyle\text{sech}^2 udu}=\displaystyle\tanh u\)
- 32.
- \(\small \displaystyle\int{\displaystyle\text{csch}^2 udu}=-\displaystyle\coth u\)
- 33.
- \(\small \displaystyle\int{\displaystyle\tanh^2 udu}=u-\displaystyle\tanh u\)
- 34.
- \(\small \displaystyle\int{\displaystyle\coth^2 udu}=u-\displaystyle\coth u\)
- 35.
- \(\small \displaystyle\int{\displaystyle\sinh^2 udu}=\displaystyle\frac{\displaystyle\sinh 2u}{4}-\displaystyle\frac{u}{2}=\displaystyle\frac{1}{2}(\displaystyle\sinh u\displaystyle\cosh u-u)\)
- 36.
- \(\small \displaystyle\int{\displaystyle\cosh^2 udu}=\displaystyle\frac{\displaystyle\sinh 2u}{4}+\displaystyle\frac{u}{2}=\displaystyle\frac{1}{2}(\displaystyle\sinh u\displaystyle\cosh u+u)\)
- 37.
- \(\small \displaystyle\int{\displaystyle\text{sech} u\displaystyle\tanh udu}=-sech u\)
- 38.
- \(\small \displaystyle\int{\displaystyle\text{csch} u\displaystyle\coth udu}=-csch u\)
- 39.
- \(\small \displaystyle\int{\displaystyle\frac{du}{u^{2}+a^{2}}}=\displaystyle\frac{1}{a}\displaystyle\tan^{-1}\displaystyle\frac{u}{a}\)
- 40.
- \(\small \displaystyle\int{\displaystyle\frac{du}{u^{2}-a^{2}}}=\displaystyle\frac{1}{2a}\ln\left(\displaystyle\frac{u-a}{u+a}\right)=-\displaystyle\frac{1}{a}\displaystyle\coth^{-1}\displaystyle\frac{u}{a}\quad;\quad u^{2}>a^{2}\)
- 41.
- \(\small \int{\frac{du}{a^{2}-u^{2}}}=\frac{1}{2a}\ln\left(\frac{a+u}{a-u}\right)=\frac{1}{a}\tanh^{-1}\frac{u}{a}\quad;\quad u^{2}< a^{2}\)
- 42.
- \(\small \displaystyle\int{\displaystyle\frac{du}{\displaystyle\sqrt{a^{2}-u^{2}}}}=\sin^{-1}\displaystyle\frac{u}{a}\)
- 43.
- \(\small \displaystyle\int \displaystyle\frac{du}{\displaystyle\sqrt{u^{2}+a^{2}}} = \displaystyle\ln \left( u+ \displaystyle\sqrt{u^{2} + a^{2}} \right)\)
- 44.
- \(\small \displaystyle\int \frac{du}{\displaystyle\sqrt{u^{2}-a^{2}}} = \displaystyle\ln \left( u + \displaystyle\sqrt{u^{2} - a^{2}} \right)\)
- 45.
- \(\small \displaystyle\int{\frac{du}{u\displaystyle\sqrt{u^{2}-a^{2}}}}=\displaystyle\frac{1}{a}sec^{-1}\left|\displaystyle\frac{u}{a}\right|\)
- 46.
- \(\small \displaystyle\int{\displaystyle\frac{du}{u\displaystyle\sqrt{u^{2}+a^{2}}}}=-\displaystyle\frac{1}{a}\displaystyle\ln\left(\displaystyle\frac{a+\displaystyle\sqrt{u^{2}+a^{2}}}{u}\right)\)
- 47.
- \(\small \displaystyle\int{\displaystyle\frac{du}{u\displaystyle\sqrt{a^{2}-u^{2}}}}=-\displaystyle\frac{1}{a}\displaystyle\ln\left(\frac{a+\displaystyle\sqrt{u^{2}-a^{2}}}{u}\right)\)
- 48.
- \(\small \displaystyle\int{f^{(n)}gdx}=f^{(n-1)}g-f^{(n-2)}g'+f^{(n-3)}g''-\cdots(-1)^{n}\displaystyle\int{fg^{(n)}dx}\)
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