# Math Help - Integration

1. ## Integration

Can someone please check that my answer for this question is correct:
Question:
$
A_n = \frac{2}{3} \int^{3}_{0} \frac{1}{2}x sin(\frac{n \Pi}{3} x) dx
$

My working out:
$
A_n = \frac{1}{2} \int^{3}_{0} x sin(\frac{n \Pi}{3} x) dx$

$= \frac{1}{2} {[x \frac{3}{n \Pi} cos (\frac{n \Pi}{3} x)]^{3}_{0} +\frac{3}{n \Pi} \int^{3}_{0} cos (\frac{n \Pi}{3} x) dx}$
$= \frac{1}{2} {[\frac{-9}{n \Pi} cos (n \Pi) - 0] + \frac{3}{n \Pi} [\frac{3}{n \Pi} sin (\frac{n \Pi}{3} x)]^{3}_{0}}$
$= \frac{9}{2 n^2 \Pi^2}sin(n \Pi) - \frac{9}{2 n \Pi} cos (n \Pi)$
$= \frac{9}{2 n \Pi} (-1)^(n+1) - frac{9}{2 n \Pi} cos (n \Pi)$

for some reason i think this looks wrong, can someone please confirm with me? thank you

2. Originally Posted by gconfused
Can someone please check that my answer for this question is correct:
Question:
$
A_n = \frac{2}{3} \int^{3}_{0} \frac{1}{2}x sin(\frac{n \Pi}{3} x) dx
$

My working out:
$
A_n = \frac{1}{2} \int^{3}_{0} x sin(\frac{n \Pi}{3} x) dx$

$= \frac{1}{2} {[x \frac{3}{n \Pi} cos (\frac{n \Pi}{3} x)]^{3}_{0} +\frac{3}{n \Pi} \int^{3}_{0} cos (\frac{n \Pi}{3} x) dx}$
$= \frac{1}{2} {[\frac{-9}{n \Pi} cos (n \Pi) - 0] + \frac{3}{n \Pi} [\frac{3}{n \Pi} sin (\frac{n \Pi}{3} x)]^{3}_{0}}$
$= \frac{9}{2 n^2 \Pi^2}sin(n \Pi) - \frac{9}{2 n \Pi} cos (n \Pi)$
$= \frac{9}{2 n \Pi} (-1)^(n+1) - frac{9}{2 n \Pi} cos (n \Pi)$

for some reason i think this looks wrong, can someone please confirm with me? thank you
second line multiply with 1/3 not 1/2
what you do in the last step
other things is wright

3. Originally Posted by Amer
second line multiply with 1/3 not 1/2
what you do in the last step
other things is wright
thank you for helping

well this is actually a differential equation question, but i just needed help with the integration part.. Umm i dont really know what i did in the last step

coz that's the way they simplified it in my lecture notes for partial differential equations, but yeahh i probably did that wrong too ><

4. ohh sorries the last line was:

$= \frac{9}{2 n \Pi} (-1)^{n+1} - \frac{9}{2 n \Pi} cos (n \Pi)
$

but i don't know if that's how you simplify it

5. Ok

suppose it was correct lats simplify it sin(n pi ) = 0 for all n natural number

$\frac{9}{2n\pi}cos(n\pi)=\begin{cases} \left(\frac{9}{2n\pi}\right) \mbox{ if } n..even \\ -\left(\frac{9}{2n\pi}\right) \mbox{ if } n..odd \\\end{cases}$