<s>[INST] Integration by u-substitution. Evaluate $$\int_0^1 x^2 \cos\left(\frac{x^3}{3}+1\right)\cos\left(\sin\left(\frac{x^3}{3}+1\right)\right) \mathrm{d}x$$
My attempt: Let $u = \dfrac{x^3}{3}+1$ so $\mathrm{d}u = x^2\mathrm{d}x$
$$\int_0^1\cos(u)\cos(u)\mathrm{d}u$$
Then I am stuck trying to integrate $\cos^2(x)$. Any help is good. Thanks.
 [/INST] Let $$u = \sin\left(\frac{x^3}{3}+1\right)$$
$$ \frac{du}{dx} = x^2\cos\left(\frac{x^3}{3}+1\right)$$
$$\int_0^1 \cos u \ du  = \left(\sin \sin\left(\frac{x^3}{3}+1\right) \right)_0^1 = \sin \sin\left(\frac{4}{3}\right) - \sin \sin\left(1\right) =0.08$$
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