Question

Evaluate the integral of (√siny) cos y dy from 3 to π/2

Answer 1

Solution.

$$\int\limits_3^{\frac{\pi} {2}}\sqrt{sin{y}}\cos{y}dy.$$

Make a replacement:

$t=\sqrt{\sin{y}}, t\geq0,$ from here $\sin{y}=t^2.$

Then $\cos{y}dy=2tdt.$

We will have

$\int\limits_3^{\frac{\pi}{2}}t\cdot 2tdt=\int\limits_3^{\frac{\pi}{2}}2t^2dt=\frac{2t^3}{3}|_3^{\frac{\pi}{2}}=\newline =\frac{2\sin^{\frac{3}{2}}y}{3}|_3^{\frac{\pi}{2}}= \frac{2\sin^{\frac{3}{2}}\frac{\pi}{2}}{3}-\frac{2\sin^{\frac{3}{2}}3}{3}=\newline =\frac{2}{3}-\frac{2\sin^{\frac{3}{2}}3}{3} \approx 0.63.$

Answer.

$\int\limits_3^{\frac{\pi} {2}}\sqrt{sin{y}}\cos{y}dy\approx 0.63.$