(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

∫ x^{2}sin x

2. Relevant equations

uv - ∫ v du

3. The attempt at a solution

u = x^{2}

du = 2x

dv = sin x

v = -cos x

step 1. x^{2}- cos x - ∫ -cos x 2x

I think -cos x * 2x becomes -2x cos x

so now we have

step 2. x^{2}- cos x - ∫ -2x cos x

which means I have to integrate by parts again. Here, concentrating on just the right hand side

u = 2x

du = 2

dv = cos x

v = sin x

step 3. 2x sin - ∫ sin x * 2

[after 10 minutes of research I've decided that I have to move that 2 to the left of the integral. That sort of helps. previously I took the antiderivative of 2.]

step 4. 2x sin + 2 -cosx

now add the left hand side part from above

step 5. x^{2}- cos x

step 6. x^{2}- cos x + 2x sin x + - 2 cosx + C

the book says the answer is

-x^{2}cos x + 2x sin x + 2 cos x + C

So I'm almost correct, I just don't understand how they got the negative on x^{2}, Also my right cos x is negative and their's is positive.

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# More integration by parts

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