Fourier integrals

Question

I'm starting with the fourier theme and I need to demonstrate a couple of integrals, but I'm coming back and I do not understand much, could you please help

1. $\int_d^{d+T}\sin(n\omega t)dt = 0$

2. $\int_d^{d+T}\sin(m\omega t)\sin(n\omega t)dt = \left\{ \begin{array}{ll} 0 \text{ for } m\neq n \\ \frac{1}{2T} \text{ for } m=m \end{array}\right.$

I would greatly appreciate your help

Answer 1

This a wonderful exercise that uses trig identities. I assume you can do the first integral. If you have any difficulty the answer is $\frac{-1}{nw} [cos(n w (d+T)) - cos(n w(d))]$The second one can be completed by using the trig identity $sin(x)^{2}= \frac{1}{2}(1 - cos(2x))$ for n=m case. Then you can use $sin(a)sin(b) = \frac{1}{2}[cos(a+b)-cos(a-b)]$ identity for the other. Hope this answers your question.

Answer 2

Hint: complex definitions of sine makes two into an elementary integral. No trig identities required.