Question

Find the integral $\int x \sin(x^2) dx$

• A. $-\frac{1}{2} cos(x^2) + C$
• B. $\frac{sin(x^2)}{2} + C$
• C. $-\frac{cos(x^2)}{2} + C$
• D. $-\frac{sin(x^2)}{2} + C$

Answer 1

Answer: (A), (C)

$-\frac{1}{2} \cos(x^2) + C$

Answer 2

The integral $\int x \sin(x^2) dx$ is solved using u-substitution. Let $u = x^2$, then $du = 2x \, dx$, which implies $x \, dx = \frac{1}{2} du$. Substituting these into the integral yields $\int \sin(u) \frac{1}{2} du$. Integrating $\frac{1}{2} \sin(u)$ with respect to $u$ gives $-\frac{1}{2} \cos(u) + C$. Finally, substitute back $u=x^2$ to get $-\frac{1}{2} \cos(x^2) + C$.