Let's calculate the integration
$$\int_{0}^{\infty}\frac{r^{n-1}}{(1+r^2)^{(n+1)/2}}dr.$$
then let $r=\tan\theta$,
according to my book, its result is
$$\int_{0}^{\pi/2}\sin^{n-1}\theta d\theta.$$
but my calculation is
$$\int_{0}^{\pi/2}\cos^{2}\theta \sin^{n-1}\theta d\theta.$$
Is my book wrong?