Evaluate the following integrals:
Question:

Evaluate the following integrals:

$\int x^{2} e^{x^{3}} \cos x^{3} d x$

Solution:

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

$\mathrm{x}^{3}=\mathrm{t}$

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

$I=\frac{1}{3} \int e^{t} \cos t d t$

We know that, $\int e^{a x} \cos b x d x=\frac{e^{2 x}}{a^{2}+b^{2}}\{a \cos b x-b \sin b x\}+c$

$I=\frac{1}{3}\left[\frac{e^{t}}{2}(\cos t+\sin t)\right]+c$

$I=\frac{1}{3}\left[\frac{e^{x^{3}}}{2}\left(\cos x^{3}+\sin x^{3}\right)\right]+c$

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