How to do this integral analytically to get the form of result showed up below?

Question

How to do this integral below analytically? $$\frac{1}{T_s}\int_{0}^{T_s}{\rm exp}(j2\pi(f_k-f_l)t){\rm d}t$$ One of the forms showed in my textbook is $${\rm exp}(j\pi(f_k-f_l)T_s)\frac{\sin (\pi(f_k-f_l)T_s)}{\pi(f_k-f_l)T_s}$$ How to get the result above analytically?

Answer 1

The important equality is $$ \sin(aT)=\frac{\exp(ajT)-\exp(-ajT)}{2j}. $$ Integrating, $$ \begin{aligned} \int_0^T\exp(2ajt)\,dt&=\Bigl[\frac{\exp(2ajt)}{2aj}\Bigr]_0^T=\frac{1}{2aj}\bigl(\exp(2ajT)-1\bigr)\\ &=\exp(ajT)\frac{\exp(ajT)-\exp(-ajT)}{2aj}\\ &=\exp(ajT)\frac{\sin(aT)}{a}. \end{aligned} $$ Now, divide by $T$ and insert your value $a=\pi(f_k-f_l)$. I also wrote $T$ instead of $T_s$.