$5)\displaystyle\int\limits^{\tfrac{\pi}{2}}_0 \cos^nx\sin x \, dx\\ =-\displaystyle\int\limits^{\tfrac{\pi}{2}}_0 \cos^nx \, d(\cos x)\\ =-\dfrac{\cos^{n+1}x}{n+1} \Big|^{\tfrac{\pi}{2}}_0\\ =\dfrac{1}{n+1}\\ 8)\displaystyle\int\limits^{\tfrac{\pi}{2}}_0 \sin^3x \, dx\\ =-\displaystyle\int\limits^{\tfrac{\pi}{2}}_0 \sin^2x \, d(\cos x)\\ =\displaystyle\int\limits^{\tfrac{\pi}{2}}_0 (\cos^2x -1) \, d(\cos x)\\ =\left(\dfrac{\cos^2x}{3}-\cos x\right)\Big|^{\tfrac{\pi}{2}}_0 \\ =\dfrac{2}{3}$
