I have been trying to find the Fourier Transform of $\cos(bt).u(t)$
I applied the definition $$\int_{-\infty}^{\infty} \cos(bt).u(t).e^{-jwt} dt = \int_{0}^{\infty} \cos(bt).e^{-jwt} dt = \int_{0}^{\infty} \frac{e^{jbt} + e^{-jbt}}{2} .e^{-jwt} dt$$ Which simplifies to $$ \frac{1}{2} \int_{0}^{\infty} e^{j(b-w)t} + e^{-j(b+w)t}dt$$
But now I'll need to compute $e^\infty$ which isn't computable
How to handle this case?