Need Help in UV rule of integrals

[fringe jasper]

Hello. I am trying to learn some basic integrals. I learnt about integration by substitution, and now I'm learning about UV rules in integrals.

I have a small doubt regarding the UV rule.

∫ x sin²(x) dx

This integral, in specific.
We could replace sin²(x) by 1 - cos(2x) / 2, and then just simply solve the integral.

But what I wanna ask is, can we apply the UV rule directly into the original question. I was told that to apply the UV rule, the function in question (here it is sin²x) must not have powers. Is this true? Can we not apply the UV rule directly here, considering sin²x as a function, or do we have to convert it first, then only we can apply the UV rule.

I understand this may be a beginners question, but so am I. Thank you.

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[terse iron]

No, that's not necessary. Here we can do the following:
u = x, dv = sin(x)^2 dx
As sin(x)^2 = (1 - cos(2x))/2, we get du = dx, v = (1/2)(x - (1/2)sin(2x)). So:
∫(x sin(x)^2 dx) = (1/2)x(x - (1/2)sin(2x)) - (1/2)∫((x - (1/2)sin(2x))dx)
Thre rest is easy.

Basically, if we have an expression like P(x) Q(e^(ax)), P(x) Q(sin(bx)) or P(x) Q(cos(bx)), where P and Q are polynomials, we can always take u = P(x) and the rest as dv.