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?