Mathematics

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


ANSWER

$$\dfrac{-\cos^2x}{2}+C$$


SOLUTION

Consider the given integral.

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

 

Let,

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

 

Therefore,

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

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

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

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

 

Hence, this is the required result.
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