evaluate the limits of pie and 0 (x+sinnx)dx

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- Feb 1st 2007, 06:26 PMlay137cal 2
evaluate the limits of pie and 0 (x+sinnx)dx

- Feb 1st 2007, 07:17 PMThePerfectHacker
What are you trying to find? The Fourier Series?

We need to find, (assume $\displaystyle n>1$)

$\displaystyle \int_0^{\pi} (x+\sin nx )dx$

By linearity,

$\displaystyle \frac{1}{2}x^2-\frac{1}{n}\cos nx\big|^{\pi}_0$

$\displaystyle \frac{1}{2}\pi^2-\frac{1}{n}\cos \pi n+\frac{1}{2}$

If $\displaystyle n$ is even, then we have,

$\displaystyle \frac{1}{2}\pi^2 - \frac{1}{2}+\frac{1}{2}=\frac{1}{2}\pi^2$.

If $\displaystyle n$ is odd, then we have,

$\displaystyle \frac{1}{2}\pi^2+\frac{1}{2}+\frac{1}{2}=\frac{1}{ 2}\pi^2+1$ - Feb 2nd 2007, 04:21 AMtopsquark