Real Variable, Complex Integral $\int_{o}^{\pi} ie^{3it}\,dt$

Question

So far I used $u$-substitution and got the work below.

$$\int_{0}^{\pi} ie^{3it} dt= \frac{1}{3}e^{3it}\Big|_0^\pi $$

Now, I just want to be sure that $e^{3it}=\cos(3t)+i\sin(3t)$ is correct so I can finish solving the rest of the problem.

However, if my method is incorrect please point me in the right direction.

Answer 1

Your working is fine so far, just evaluate

$$\frac13 \left( \cos(3t) +i \sin(3t) \right) \mid_0^\pi$$