What is the integral of e^{iθ} ?

The integral of e^{iθ} is sin(θ)-icos(θ)+C

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integral of e^{iθ}

\int e^{i θ} d θ

sin (θ) - i cos (θ) + C

Solution steps

\int e^{i θ} d θ

Apply u-substitution

= \int - i e^{u} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= - i \cdot \int e^{u} d u

Use the common integral:

\int e^{u} d u = e^{u}

= - i e^{u}

Substitute back

u = i θ

= - i e^{i θ}

Simplify

- i e^{i θ} : sin (θ) - i cos (θ)

= sin (θ) - i cos (θ)

Add a constant to the solution

= sin (θ) - i cos (θ) + C

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