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

The integral from-pi to pi of xe^{-inx} is (2pii(-1)^n)/n

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

integral from-pi to pi of xe^{-inx}

\int_{- π}^{π} x e^{- in x} d x

\frac{2 πi ( - 1 ) ^{n}}{n}

Solution steps

\int_{- π}^{π} x e^{- in x} d x

Simplify

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

= \int_{- π}^{π} 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_{- π}^{π} x cos (n x) d x - \int_{- π}^{π} i x sin (n x) d x

\int_{- π}^{π} x cos (n x) d x = 0

\int_{- π}^{π} i x sin (n x) d x = - \frac{2 πi ( - 1 ) ^{n}}{n}

= 0 - (- \frac{2 πi ( - 1 ) ^{n}}{n})

Simplify

= \frac{2 πi ( - 1 ) ^{n}}{n}

What is the integral from-pi to pi of xe^{-inx} ?
The integral from-pi to pi of xe^{-inx} is (2pii(-1)^n)/n