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Popular >

integral of e^{-ix}

Solution

\int e^{- i x} d x

Solution

sin (x) + i cos (x) + C

Solution steps

\int e^{- i x} d x

Simplify

e^{- i x} : cos (x) - i sin (x)

= \int cos (x) - i sin (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 (x) d x - \int i sin (x) d x

\int cos (x) d x = sin (x)

\int i sin (x) d x = - i cos (x)

= sin (x) - (- i cos (x))

Simplify

= sin (x) + i cos (x)

Add a constant to the solution

= sin (x) + i cos (x) + C

Popular Examples

limit as x approaches+1 of |2x|

x \to + 1 lim (∣ 2 x ∣)

derivative of (x^2-4/(6x^3-2))

\frac{d}{d x} (\frac{x ^{2} - 4}{6 x ^{3} - 2})

derivative of 1/(2sqrt(x))+1/(4x^{3/4)}derivative of

\frac{1}{2 x} + \frac{1}{4 x ^{\frac{3}{4}}}

derivative of (x^2-2/(-x^2+1))

\frac{d}{d x} (\frac{x ^{2} - 2}{- x ^{2} + 1})

derivative of arcsin(2x-1)

\frac{d}{d x} (arcsin (2 x - 1))

Frequently Asked Questions (FAQ)

What is the integral of e^{-ix} ?
The integral of e^{-ix} is sin(x)+icos(x)+C