How to integrate e$^{int}$ with respect to $t$?

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I am wanting to show that $\int_{0}^{2\pi}e^{nit}dt=0$ for $n\neq0$, but I am unsure if it is correct to write $$\int_{0}^{2\pi}e^{int}\mathrm dt=\left.\frac{e^{int}}{ni}\right|_{0}^{2\pi}$$ or $$\int_{0}^{2\pi}e^{int}\mathrm dt=\left.\frac{e^{int}}{n}\right|_{0}^{2\pi}$$

I know both will yield the same answer but it will help to know for future reference.

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3 Answers

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The first one is correct; the second one is not. The formula arises from the fact that $\frac d {dt} e^{int}=ine^{int}$.

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Set $ni=k$, then you have: $$\int_0^{2\pi}e^{kt}dt=\bigg(\frac{e^{kt}}{k}\bigg)^{2\pi}_0$$

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The first is correct; $\mathrm i$ is a constant too (so it has to be in the antiderivative).

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