integrate 2

\int (xsecx)\frac{x.cosx}{(xsinx+cosx)^2}dx

take x.sec(x) as first function and \frac{x.cosx}{(xsinx+cosx)^2} as second function and then integrate by parts

\frac{x^{2}}{(xsinx+cosx)^{2}}=

here (xsinx+cosx)=\sqrt{1+x^{2}}(\frac{x}{\sqrt{1+x^{2}}}sinx+\frac{1}{\sqrt{1+x^{2}}}cosx)=\sqrt{1+x^{2}}(sin\alpha sinx+cos\alpha cosx)=\sqrt{1+x^{2}}(cos(x-\alpha ))

(xsinx+cosx)^{2}=(1+x^{2})cos^{2}(x-\alpha )
here sin\alpha =\frac{x}{\sqrt{1+x^{2}}} and cos\alpha =\frac{1}{\sqrt{1+x^{2}}}
put these value to integral
\int \frac{x^{2}}{(1+x^{2})}sec^{2}(x-\alpha )dx=% 5Calpha =tan^{-1}(x)
and then put (x-\alpha )=t
here \alpha =tan^{-1}(x)