Exercise:
Compute, ∫π0xsinx1+cos2xdx
Solution: Let I=∫π0xsinx1+cos2xdx
So I=∫π0(π−x)sin(π−x)1+cos2(π−x)dx=∫π0(π−x)sinx1+(−cosx)2dx=∫π0πsinx−xsinx1+cos2xdx
⇒∫π0xsinx1+cos2xdx=∫π0πsinx1+cos2xdx−∫π0xsinx1+cos2xdx
⇒2∫π0xsinx1+cos2xdx=∫π0πsinx1+cos2xdx
Let u=cosx⇔du=−sinxdx
So ∫π0πsinx1+cos2xdx=−π∫u(π)u(0)du1+u2=−π∫−11du1+u2=π∫1−1du1+u2
=πarctan(u)1−1=π(arctan(1)−arctan(−1))=π(π4+π4)=π(π2)=π22
Hence ∫π0xsinx1+cos2xdx=π24
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