<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] $$\int_0^1 x^2 \cos\left(\frac{x^3}{3}+1\right)\cos\left(\sin\left(\frac{x^3}{3}+1\right)\right) dx$$
Let $u=\sin\left(\frac{x^3}{3}+1\right)$, then
$$du=x^2\cos\left(\frac{x^3}{3}+1\right)dx$$
So now
$$\int_{\sin 1}^{\sin\frac43} \cos u\ du=\sin\left(\sin \frac43 \right)-\sin\left(\sin 1\right)$$
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