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Integral exercise depend on baf(x)dx=baf(a+bx)dx


Exercise:

Compute, π0xsinx1+cos2xdx

Solution: Let I=π0xsinx1+cos2xdx

So I=π0(πx)sin(πx)1+cos2(πx)dx=π0(πx)sinx1+(cosx)2dx=π0πsinxxsinx1+cos2xdx

π0xsinx1+cos2xdx=π0πsinx1+cos2xdxπ0xsinx1+cos2xdx

2π0xsinx1+cos2xdx=π0πsinx1+cos2xdx

Let u=cosxdu=sinxdx

So π0πsinx1+cos2xdx=πu(π)u(0)du1+u2=π11du1+u2=π11du1+u2

=πarctan(u)11=π(arctan(1)arctan(1))=π(π4+π4)=π(π2)=π22

Hence π0xsinx1+cos2xdx=π24

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