Substitution Rule for Indefinite Integrals — Question 4

Question 4 - Solution

Use substitution. Let $$u = \cos(x^2).$$ Then $$du = -\sin(x^2)\cdot 2x\,dx \quad\Rightarrow\quad -\,\frac{1}{2}\,du = x\sin(x^2)\,dx.$$

Substitute into the integral: $$\int \frac{x\,\sin(x^2)}{\sqrt{1+\cos(x^2)}}\,dx = -\frac{1}{2}\int \frac{du}{\sqrt{1+u}}.$$

Integrate: $$-\frac{1}{2}\int (1+u)^{-1/2}\,du = -\frac{1}{2}\cdot 2(1+u)^{1/2} = -\sqrt{1+u}.$$

Substitute back: $$\boxed{ -\sqrt{1+\cos(x^2)} + C }$$