Question

Find the Fourier expansion for f(x)=4x, x[-pi,pi], T=2 pi

Answer 1

This is the expansion of f(x)=x
You can multiply by 4 to meet yours.

Answer 2

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Answer 3

@3mar is awesome, he can help you :D

Answer 4

f(x) = 4x is odd so you only need the sine terms of the expansion

So coefficients are:

\(b_n = \dfrac{1}{\pi}\int\limits_{-\pi}^{\pi} 4x \, \sin nx\, dx\)

\( = \dfrac{8}{\pi}\int\limits_{0}^{\pi} x \dfrac{d}{dx}(- \dfrac{1}{n}\cos(nx) ) \, dx\)

ie do it by IBP

Answer 5

Thanks for the medal!
Hope I could help you.