Science:Math Exam Resources/Courses/MATH101 C/April 2025/Question 02 (a)/Solution 1

We consider the following $u$-substitution: $u=\sin x,\mathrm{d}u=\cos x\mathrm{d}\mathrm{x}$. The integral then becomes

$$\int \cos x\sqrt{\sin x}\mathrm{d}x=\int u^{1/2}\mathrm{d}u=\frac{2}{3}{u}^{3/2}+C.$$

Substituting $u=\sin x$ back we get:

$$\int \cos x\sqrt{\sin x}\mathrm{d}x=\frac{2}{3}{\sin }^{}(x)+C.$$