1. Integration by parts.

I almost understand all of this question, except in the last step where we have "128". I will show my work up to that step so you can follow.

$\displaystyle \int{x^2sin\frac{x}{4}}dx$

$\displaystyle -4x^2cos\frac{1}{4}x--8\int{xcos\frac{1}{4}}xdx$

$\displaystyle -4x^2cos\frac{1}{4}x+8[{4xsin\frac{1}{4}x-4\int{sin\frac{1}{4}xdx}}]$

$\displaystyle -4x^2cos\frac{1}{4}x+32xsin\frac{1}{4}x+128cos\frac {1}{4}x+c$

I thought that distributing the 8 in step 3 would give us "+32cos...x+c?"

2. Originally Posted by gammaman
I almost understand all of this question, except in the last step where we have "128". I will show my work up to that step so you can follow.

$\displaystyle \int{x^2sin\frac{x}{4}}dx$

$\displaystyle -4x^2cos\frac{1}{4}x--8\int{xcos\frac{1}{4}}xdx$

$\displaystyle -4x^2cos\frac{1}{4}x+8[{4xsin\frac{1}{4}x-4\int{sin\frac{1}{4}xdx}}]$

$\displaystyle -4x^2cos\frac{1}{4}x+32xsin\frac{1}{4}x+128cos\frac {1}{4}x+c$

I thought that distributing the 8 in step 3 would give us "+32cos...x+c?"
It comes from dividing the 32 by the 1/4 in sin(x/4) to give 128

3. Ok I see it now. The anti-derivative of sin[(1/4)] is -4cos so -4 x -4 = +16
16 x the 8 in front gives us 128.