<s>[INST] Calculus integration problem: $\int \sin^5 (x) \cos^2 (x)\,dx$ What's the integration of $$\int \sin^5 (x) \cos^2 (x)\,dx?$$
 [/INST] $$ \int \sin^5 (x) \cos^2(x) dx $$
$$= \int(\sin^2(x))^2 \cos^2(x) \sin(x) dx$$ 
$$=-\int(1 - \cos^2(x))^2 cos^2(x) (-sin(x) dx) $$
Let $u = \cos(x)$ 
$\implies du = -\sin(x) dx$ 
$$= -\int(1 - u^2)² u² (du)$$ 
$$= -\int(1 - 2u^2 + u^4) u^2 du $$
$$= -\int(u^2 - 2u^4+ u^6) du$$ 
$$= -\left(\frac{u^3}{3} - \frac{2u^5}{5} + \frac{u^7}{7}\right) + C$$ 
$$= -u^3\left(\frac{1}{3} - \frac{2u^2}{5} +\frac{ u^4}{7}\right) + C $$
$$= -\cos^3(x) \left(\frac{1}{3} - \frac{2\cos^2(x)}{5} + \frac{\cos^4(x)}{7}\right) + C $$
$$= -\cos^3(x)\frac{15\cos^4(x) - 42\cos^2(x) + 35}{105} + C $$
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