Substitution Rule for Indefinite Integrals — Question 9

Question 9

Evaluate the integral $$\int \frac{\cos(\sqrt{x})}{\sqrt{x}}\,dx.$$

Question 9 - Solution

Rewrite the integrand to reveal a derivative pattern: $$2\cos(\sqrt{x})\cdot \frac{d}{dx}(\sqrt{x}).$$

Let $$u=\sqrt{x}.$$ Then $$du=\frac{1}{2\sqrt{x}}\,dx \quad\Rightarrow\quad dx=2\sqrt{x}\,du.$$

Substitute into the integral: $$\int \frac{\cos(\sqrt{x})}{\sqrt{x}}\,dx = \int \frac{\cos(u)}{\sqrt{x}}\cdot 2\sqrt{x}\,du = 2\int \cos u\,du.$$

Integrate: $$2\int \cos u\,du = 2\sin u.$$

Substitute back: $$\boxed{ 2\sin(\sqrt{x}) + C }$$