Combining integrals with swapped limits

jinksys · Dec 4, 2011

! SOLVED !

This isn't a homework problem, just an equation in my chapter. I don't see how the two integrals pointed to by the blue arrows become the integral pointed to by the red arrow. I know that if you swap the limits of integration, you change the sign of the integral. However, how do you change the -pi to pi in the second integral?

Printed textbook page showing the Fourier coefficient a_k computed as a sum of two integrals, with an integral from 0 to pi of e^(-ikx) plus an integral from -pi to 0 of (-1)e^(-ikx), followed by the combined integral of (e^(-ikx) - e^(ikx)) over 0 to pi and then -i/pi times the integral of sin kx.

hunt_mat · Dec 4, 2011

Use the substitution u=-x

jinksys · Dec 4, 2011

Of course, thanks.

Combining integrals with swapped limits

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