User:Sree2lucky

\documentclass{article} \begin{document} \begin{enumerate} %this is created by sredhar% \item$\displaystyle\int af(u)d(u)=\ a\int f(u) d(u)$ \item$\displaystyle\int \left( \sum_{i=1}^n\,a_{i}f_{i}(u)\right)du= \sum_{i=1}^n \left( \int a_{i} f_{i}(u)du\right)$ \item$\displaystyle\int u^n du =\frac{u^{n+1}}{n+1}+c\, n\neq -1$ \item$\displaystyle\int u^-1 du=ln|u|+c$ \item$\displaystyle\int e^{bu}du=\frac{a^{bu}}{blna}+c,a>0,a\neq1$ \item$\displaystyle\ln|u|du=uln|u|-u+c$ \item$\displaystyle\int sin(au)du=\frac{-cos(au)}{a}+c$ \item$\displaystyle\int cos(au)du=\frac{sin(au)}{a}+c$ \item$\displaystyle\int tan(au)du=\frac{ln|sin(au)|}{a}+c$ \item$\displaystyle\int cot(audu=\frac{ln|sin(au)|}{a}+c$ \item$\displaystyle\int sec(au)du=\frac{ln|sec(au)+tan(au)|}{a}+c$ \item$\displaystyle\int csc(au)du=\frac{ln|csc(au)-cot(au)|}{a}+c$ \item$\displaystyle\int sinh(au)du=\frac{cosh(au)}{a}+c$ \item$\displaystyle\int cosh(au)du=\frac{sinh(au)}{a}+c$ \item$\displaystyle\int tanh(au)du=\frac{ln|cosh(au)|}{a}+c$ \item$\displaystyle\int coth(au)du=\frac{ln|sinh(au)|}{a}+c$ \item$\displaystyle\int sech(au)du=\frac{2}{a}arctan(e{au})+c$ \item$\displaystyle\int csch(au)du=\frac{2}{a}arctan(e^{au})+c$ \item$\displaystyle\int sin^2(au)du=\frac{u}{2}-\frac{sin(au)cos(au)}{2a}+c$ \item$\displaystyle\int cos^2(au)du=\frac{u}{2}+\frac{sin(au)cos(au)}{2a}+c$ \item$\displaystyle\int tan^2(au)du=\frac{tan(au)}{a}-u+c$ \item$\displaystyle\int cot^2(au)du=-\frac{cot(au)}{a}-u+c$ \item$\displaystyle\int sec^2(au)cu=\frac{tan(au)}{a}+c$ \item$\displaystyle\int csc^2(au)du=-\frac{-cot(au)}{a}+c$ \item$\displaystyle\int sinh^2(au)du=\frac{-u}{2}+\frac{sinh(2au)}{4a}+c$ \item$\displaystyle\int cosh^2(au)du=\frac{u}{2}+\frac{sinh(2au)}{4a}+c$ \item$\displaystyle\int tanh^2(au)du=u-\frac{tanh(au)}{a}+c$ \item$\displaystyle\int cot^2(au)du=u-\frac{coth(au)}{a}+c$ \item$\displaystyle\int sech^2(au)du=\frac{tanh(au)}{a}+c$ \item$\displaystyle\int csch^2(au)du=\frac{-coth(au)}{a}+c$ \item$\displaystyle\int sec(au)tan(au)du=\frac{sec(au)}{a}+c$ \item$\displaystyle\int csc(au)cot(au)du=\frac{-csc(au)}{a}+c$ \item$\displaystyle\int sech(au)coth(au)du=\frac{-csch(au)}{a}+c$ \item$\displaystyle\int csch(au)coth(au)du=\frac{-csc(au)}{a}+c$ \item$\displaystyle\int \frac{du}{a^2+u^2}=\frac{1}{a}arctanh \left(\frac{u}{a}\right)+c$ \item$\displaystyle\int \frac{du}{a^2-u^2}=\frac{1}{a}arctanh \left(\frac{u}{a}\right) +c=\frac{1}{2a}ln| \frac{a+u}{a-u}|+c$ \item$\displaystyle\int \frac{du}{\sqrt {a^2+u^2}}=arcsinh \left(\frac{u}{a} \right)+c$ \item$\displaystyle\int \frac{du}{\sqrt {a^2-u^2}}=arcsin \left(\frac{u}{a} \right)+c,|a|>|u|$ \item$\displaystyle\int \frac{du}{\sqrt {u^2-a^2}}=arccosh \left(\frac{u}{a} \right) +c, |u|>|a|$ \item$\displaystyle\int \frac{du}{u \sqrt {u^2-a^2}}=\frac{1}{a}arcsech \left(\frac{u}{a} \right)+c, |u|>|a|$ \item$\displaystyle\int \frac{du}{u \sqrt {a^2+u^2}}=\frac{-1}{a}arcsech \left(\frac{u}{a}\right)+c,|a|>|u|$ \item$\displaystyle\int \frac{du}{u \sqrt {a^2+u^2}}=\frac{-1}{a}arccsch \left(\frac{u}{a} \right) +c$ \item$\displaystyle\int \frac{u du}{\sqrt a^2+u^2}=\sqrt{a^2+u^2}+c$ \item$\displaystyle\int \frac{u du}{\sqrt {a^2+u^2}}+c$ \item$\displaystyle\int \frac{u du}{a^2-u^2}=-ln \sqrt {a^2-u^2}+c, |a|>|u|$ \item$\displaystyle\int \frac{u du}{\sqrt {a^2+u^2}}=\sqrt {a^2+u^2}+c$ \item$\displaystyle\int \frac{u du} {\sqrt {a^2-u^2}}= - \sqrt {a^2-u^2}+c, |a|> |u|$ \item$\displaystyle\int \frac{u du}{\sqrt {u^2-a^2}}=\sqrt {u^2-a^2}+c, |u|>|a|$ \item$\displaystyle\int arcsin(au)du=u arcsin(au)+\frac{1}{a}\sqrt{1-a^2u^2}+c,|a||u|<1$ \item$\displaystyle\int arccos(au)du=u arccos(au) \frac{-1}{a}\sqrt {1-a^2u^2}+c, |a||u|<1$ \item$\displaystyle\int arctan(au)du=u arctan(au)\frac{-1}{2a}ln(1+a^2u^2)+c$ \item$\displaystyle\int arccot(au)du=uarccot(au)+\frac{1}{2a}ln(1+a^2u^2)+c$ \item$\displaystyle\int arcsec(au)du=uarcsec(au)\frac{-1}{a}ln|au+ \sqrt {a^2u^2-1}|+c,au>1$ \item$\displaystyle\int arccsc(au)du=uarccsc(au)+\frac{1}{a}ln|au+\sqrt {a^2u^2-1}|+c, au>1$ \item$\displaystyle\int arcsinh(au)du=uarcsinh(au)=uarcsinh(au)\frac{-1}{u}\sqrt {1+a^2u^2}+c$ \item$\displaystyle\int arccosh(au)du=uarccosh(au)-\frac{-1}{a}\sqrt {-1+a^2u^2}+c, |a||u|>1$ \item$\displaystyle\int arctanh(au)du=uarctanh(au)+\frac{1}{2a}ln(-1+a^2u^2)+c, |a||u| \neq1$ \item$\displaystyle\int arccoth(au)du=uarccoth(au)+\frac{1}{2a}ln(-1+a^2u^2)+c,|a||u| \neq1$ \item$\displaystyle\int arcsech(au)du=uarcsech(au)+\frac{1}{a}arcsin(au)+c,|a||u|<1$ \item$\displaystyle\int arccsch(au)du=uarccsch(au)+\frac{1}{a}ln|au+\sqrt {a^2u^2+1}|+c$ \item$\displaystyle\int e^{au}sin(bu)du=\frac{e^{au} \left[asin(bu)-bcos(bu)\right] }{a^2+b^2}+c$ \item$\displaystyle\int e^{au}cos(bu)du=\frac{e^{au}\left[acos(bu)+bsin(bu)\right] }{a^2+b^2}+c$ \item$\displaystyle\int sin^n(u)du=\frac{-1}{n}\left[sin^{n-1}(u)cos(u)\right]+\frac{n-1}{n}\int sin^{n-2}(u)du$ \item$\displaystyle\int cos^n(u)du=\frac{1}{n}\left[cos^{n-1}(u)sin(au)\right]+\frac{n-1}{n}\int cos^{n-2}(u)du$ \item$\displaystyle\int tan^n(u)du=\frac{tan^{n-1}(u)}{n-1}-\int cot^{n-2}(u)du$ \item$\displaystyle\int cot^n(u)du=\frac{-cot^{n-1}(u)}{n-1}-\int cot^{n-2}(u)du$ \item$\displaystyle\int sec^n(u)du=\frac{1}{n-1}\left[sec^{n-2}(u)tan(u)\right]+\frac{n-2}{n-1}\int sec^{n-2}(u)du$ \item$\displaystyle\int csc^n(u)du=\frac{-1}{n-1}\left[csc^{n-2}(u)cot(u)\right]+\frac{n-2}{n-1}\int csc^{n-2}(u)du$ \item$\displaystyle\int sin(mu)sin(nu)du=\frac{sin \left[(m-n)u \right] }{2(m-n)}-\frac{sin \left[(m+n)u \right] }{2(m+n)}+c, m^2\neq n^2$ \item$\displaystyle\int cos(mu)cos(nu)du=\frac{sin \left[(m+n)u \right] }{2(m+n)}+c,m^2\neq n^2$ \item$\displaystyle\int sin(mu)cos(nu)du=\frac{cos \left[(m-n)u \right] }{2(m-n)}-\frac{cos \left[(m+n)u \right] }{2(m+n)}+c,m^2\neq n^2$ \end{enumerate} \end{document}