- #1

- 62

- 0

## Homework Statement

I want to integrate [tex]\int_{0}^{a} xsin\frac{\pi x}{a}sin\frac{\pi x}{a}dx[/tex]

## Homework Equations

I have the orthogonality relation:

[tex]\int_{0}^{a} sin\frac{n\pi x}{a}sin\frac{m\pi x}{a}dx = \begin{cases} \frac{a}{2} &\mbox{if } n = m; \\

0 & \mbox{otherwise.} \end{cases} [/tex]

and the parts formula:

[tex]\int u \, dv=uv-\int v \, du[/tex]

## The Attempt at a Solution

I took [itex]u = x[/itex], and [itex] dv = sin\frac{n\pi x}{a}sin\frac{m\pi x}{a}[/itex]. Following the parts formula I get a final answer of [itex]\frac{a^{2}}{2} - \frac{a^{2}}{2} = 0[/itex]. However, this is incorrect. The correct answer for the integral is [itex]\frac{a^{2}}{4}[/itex].

I know how to do this using a trigonometric identity to swap out the [itex]sin^{2}x[/itex] term for a linear term involving [itex]cos2x[/itex], but I don't quite understand why this method doesn't work. I think it has something to do with linearity but I don't fully understand why one method works and the other doesn't.

Last edited: