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# core 4 - integration by parts watch

1. ∫xsin0.5x dx with limits 0.5pi and zero

u=x
du/dx=1

v=-0.5cos0.5x
dv/dx=sin0.5x

=uv-∫du/dxv
=(x)(-1/2cos0.5x)-∫-0.5cos0.5x
=(x)(-1/2cos0.5x)+0.25sin0.5x
applying the limits to this ^ gives me:
(-1/4pi(√2/2)+1/4(√2/2)+1/2

However the answer in the text book states:
0.5√2(4-π)??
2. (Original post by Chelsea12345)
∫xsin0.5x dx with limits 0.5pi and zero

u=x
du/dx=1

v=-0.5cos0.5x
dv/dx=sin0.5x

=uv-∫du/dxv
=(x)(-1/2cos0.5x)-∫-0.5cos0.5x
=(x)(-1/2cos0.5x)+0.25sin0.5x
applying the limits to this ^ gives me:
(-1/4pi(√2/2)+1/4(√2/2)+1/2

However the answer in the text book states:
0.5√2(4-π)??
you integrates sin(0.5x) wrong you should get -2cos(0.5x)
3. The textbook is correct. You made a mistake integrating.

u=x --> du=dx
v = integral sin(0.5x) dx = -2cos(0.5x)

So uv-integral v du is...?

I think you're confusing derivative rules with integral ones.
4. using the right integral this time, i get :
=-2xcos0.5x +4sin0.5x
applying the limits to this^
-√2/2pi +2√2 -1, is this just a simpified version of the textbook answer or have i made a mistake somewhere when applying the limits?
5. (Original post by Chelsea12345)
using the right integral this time, i get :
=-2xcos0.5x +4sin0.5x
applying the limits to this^
-√2/2pi +2√2 -1, is this just a simpified version of the textbook answer or have i made a mistake somewhere when applying the limits?
F(pi/2) - F(0) whereby F(x) is your integral simplifies to -2(pi/2)(root 2/2) + 4(root 2/2) = -pi root 2 / 2 + 2 root 2 = 2 root 2 - pi root 2 / 2 = root 2/2 [4-pi].
6. (Original post by thekidwhogames)
F(pi/2) - F(0) whereby F(x) is your integral simplifies to -2(pi/2)(root 2/2) + 4(root 2/2) = -pi root 2 / 2 + 2 root 2 = 2 root 2 - pi root 2 / 2 = root 2/2 [4-pi].
Thankyou!
7. (Original post by Chelsea12345)
Thankyou!
No problem.

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