Mathematics

Integrate with respect to $$x$$:
$$2x^{2}e^{x^{2}}$$


SOLUTION

Consider the given integral.

$$I=\int{2{{x}^{2}}{{e}^{{{x}^{3}}}}}dx$$

$$I=2\int{{{x}^{2}}{{e}^{{{x}^{3}}}}}dx$$

 

Let $$t={{x}^{3}}$$

$$ \dfrac{dt}{dx}=3{{x}^{2}} $$

$$ \dfrac{dt}{3}={{x}^{2}}dx $$

 

Therefore,

$$ I=\dfrac{2}{3}\int{{{e}^{t}}}dt $$

$$ I=\dfrac{2}{3}\left( {{e}^{t}} \right)+C $$

 

Thus,

$$I=\dfrac{2}{3}\left( {{e}^{{{x}^{3}}}} \right)+C$$

 

Hence, this is the answer.

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