What is the integral of xe^{-inx} ?

The integral of xe^{-inx} is 1/(n^2)(cos(nx)+nxsin(nx))+i 1/(n^2)(-sin(nx)+nxcos(nx))+C

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integral of xe^{-inx}

\int x e^{- in x} d x

\frac{1}{n ^{2}} (cos (n x) + n x sin (n x)) + i \frac{1}{n ^{2}} (- sin (n x) + n x cos (n x)) + C

Solution steps

\int x e^{- in x} d x

Simplify

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

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

\int x cos (n x) d x = \frac{1}{n ^{2}} (n x sin (n x) + cos (n x))

\int i x sin (n x) d x = i \frac{1}{n ^{2}} (sin (n x) - n x cos (n x))

= \frac{1}{n ^{2}} (n x sin (n x) + cos (n x)) - i \frac{1}{n ^{2}} (sin (n x) - n x cos (n x))

Simplify

\frac{1}{n ^{2}} (n x sin (n x) + cos (n x)) - i \frac{1}{n ^{2}} (sin (n x) - n x cos (n x)) : \frac{1}{n ^{2}} (cos (n x) + n x sin (n x)) + i \frac{1}{n ^{2}} (- sin (n x) + n x cos (n x))

= \frac{1}{n ^{2}} (cos (n x) + n x sin (n x)) + i \frac{1}{n ^{2}} (- sin (n x) + n x cos (n x))

Add a constant to the solution

= \frac{1}{n ^{2}} (cos (n x) + n x sin (n x)) + i \frac{1}{n ^{2}} (- sin (n x) + n x cos (n x)) + C

What is the integral of xe^{-inx} ?
The integral of xe^{-inx} is 1/(n^2)(cos(nx)+nxsin(nx))+i 1/(n^2)(-sin(nx)+nxcos(nx))+C