What is the integral of xe^{ix} ?

The integral of xe^{ix} is xsin(x)+cos(x)+i(sin(x)-xcos(x))+C

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integral of xe^{ix}

\int x e^{i x} d x

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

Solution steps

\int x e^{i x} d x

Simplify

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

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

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

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

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

Add a constant to the solution

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

What is the integral of xe^{ix} ?
The integral of xe^{ix} is xsin(x)+cos(x)+i(sin(x)-xcos(x))+C