Let $\displaystyle \theta(x,\lambda)$ be the solution of

$\displaystyle \theta''+\lambda\theta=0$

$\displaystyle \theta(0)=1$

$\displaystyle \theta'(0)=0$

Use Green's formula to evaluate

$\displaystyle \displaystyle\int_0^L\theta^2(x,\lambda) \ dx$

and by specializing the result evaluate

$\displaystyle \displaystyle\int_0^L\cos^2\left(\frac{n\pi x}{L}\right) \ dx, \ \ \int_0^L\cos^2\left[\left(n+\frac{1}{2}\right)\frac{\pi x}{L}\right] \ dx$

where n is an integer.

Help!

Using Green's Formula, I obtain.

$\displaystyle \displaystyle\int_0^L\theta^2(x,\lambda) \ dx=\int_0^L[(A\theta)\bar{\theta}-\theta(A\bar{\theta}] \ dx=\theta'\bar{\theta}-\theta\bar{\theta}']_0^L=\theta'(L)\bar{\theta}(L)-\theta(L)\bar{\theta}'(L)$

where $\displaystyle \displaystyle A=-\frac{d^2\theta}{dx^2}$

Now what?