What is the integral of e^{inx} ?

The integral of e^{inx} is 1/n sin(nx)-i 1/n cos(nx)+C

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integral of e^{inx}

\int e^{in x} d x

\frac{1}{n} sin (n x) - i \frac{1}{n} cos (n x) + C

Solution steps

\int e^{in x} d x

Apply imaginary number rule:

e^{ia} = cos (a) + i sin (a)

= \int cos (n x) + i sin (n x) d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int cos (n x) d x + \int i sin (n x) d x

\int cos (n x) d x = \frac{1}{n} sin (n x)

\int i sin (n x) d x = - i \frac{1}{n} cos (n x)

= \frac{1}{n} sin (n x) - i \frac{1}{n} cos (n x)

Add a constant to the solution

= \frac{1}{n} sin (n x) - i \frac{1}{n} cos (n x) + C

\int - y e^{- y} d y

\int \frac{5}{1 + x ^{2}} d x

\frac{d}{d x} (\frac{x}{8 + x})

n = 0 \sum \infty 10 (\frac{x}{3})^{n}

x \to + 2 lim (\frac{2 - x}{x ^{2} - 4})

What is the integral of e^{inx} ?
The integral of e^{inx} is 1/n sin(nx)-i 1/n cos(nx)+C