Question

Integrate with respect to $x$:
$x\sin x^2$.

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

Answer 1

The correct option is B $\frac{-{\cos }^{2}x}{2}+C$

Consider the given integral.

$I=\int x\sin x^{2}dx$

Let,

$t=x^2\Rightarrow xdx=\frac{dt}{2}$

Therefore,

$I=\int x\sin x^{2}dx$

$I=\frac{1}{2}\int \sin tdt$

$I=-\frac{1}{2}\cos t+C$

$I=-\frac{1}{2}\cos \left(x^2\right)+C$

Hence, this is the required result.