$\displaystyle \int_{0}^{L}xSin^2\frac{n\pi x}{L} dx $ Do I have to use only integration by parts to integrate this or are there any other methods as well?
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That's what I would use.
$\displaystyle \int xsin^{2}xdx=\int \frac{1}{2}x(1-cos2x)dx$ $\displaystyle \int xcosxdx = \frac{xsin2x}{2}-\int sin2xdx$
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