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

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Solution

Consider the given integral.

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

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

Let $t = x^{3}$

$\frac{d t}{d x} = 3 x^{2}$

$\frac{d t}{3} = x^{2} d x$

Therefore,

$I = \frac{2}{3} \int e^{t} d t$

$I = \frac{2}{3} (e^{t}) + C$

Thus,

$I = \frac{2}{3} (e^{x^{3}}) + C$

Hence, this is the answer.

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