basic integration trouble...

[dangsy]

1. The problem statement, all variables and given/known data
Determine the probability of finding a particle of mass m between x=0 and x=L/10, if it is in n=3 state of an infinite well.


2. Relevant equations
P = \int_a^b\left |\psi\left(x\right)\right|^2 dx
\left|\psi\right|^2 = \frac{2}{L}Sin^2\left(\frac{nx\pi}{L}\right)


3. The attempt at a solution
I'm trying to integrate...

\int_0^{\frac{L}{10}}\frac 2 L \sin^{2}\left(\frac{3x\pi}{L}\right)dx



step (1)
\frac{1}{L}\left[ \int_0^\frac{L}{10}1-\int_0^\frac{L}{10}Cos\left(\frac{6x\pi}{L}\right) \right]dx


step (2)

\frac{1}{L}\left[\frac{L}{10} - Sin\left(\frac{3\pi}{5}\right)\right]

When I finish solving, I end up with an L in the answer...
which I know I'm not suppose to have, did I mess up my integration somewhere?
sorry about my pooooor pooor latex

[rocomath]

\int_0^{\frac{L}{10}}\frac 2 L \sin^{2}\left(\frac{3x\pi}{L}\right)dx

Yes?

[dangsy]

yep that's right sorry =(

[rocomath]

Sorry, PF was like broke last night. Use this trig identity:

\cos{2x}=\cos^{2}x-\sin^{2}x

\sin^{2}x=\frac 1 2(1+\cos{2x})

But I think that's what you did, so it's good? Sorry don't have time to work it out myself.

[dangsy]

Yes thank you!