Question:
Evaluate the following integrals:
$\int x^{2} e^{x^{3}} \cos \left(e^{x^{3}}\right) d x$
Solution:
Assume $e^{x^{3}}=t$
$\Rightarrow \mathrm{d}\left(\mathrm{e}^{\mathrm{x}^{3}}\right)=\mathrm{dt}$
$\Rightarrow 3 \mathrm{x}^{2} \cdot \mathrm{e}^{\mathrm{x}^{3}} \mathrm{dx}=\mathrm{dt}$
$\Rightarrow \mathrm{x}^{2} \cdot \mathrm{e}^{\mathrm{x}^{3}} \mathrm{dx}=\frac{\mathrm{dt}}{3}$
Substituting $\mathrm{t}$ and $\mathrm{dt}$
$\Rightarrow \int \frac{1}{3} \cos t . \mathrm{dt}$
$\Rightarrow \frac{1}{3} \sin t+c$
But $t=e^{x^{3}}$
$\Rightarrow \frac{1}{3} \sin \mathrm{e}^{\mathrm{x}^{3}}+\mathrm{c}$
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