What is the integral from 0 to 2pi of xe^{-ix} ?

The integral from 0 to 2pi of xe^{-ix} is 2ipi

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

integral from 0 to 2pi of xe^{-ix}

\int_{0}^{2 π} x e^{- i x} d x

2 iπ

Solution steps

\int_{0}^{2 π} x e^{- i x} d x

Simplify

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

= \int_{0}^{2 π} 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_{0}^{2 π} x cos (x) d x - \int_{0}^{2 π} i x sin (x) d x

\int_{0}^{2 π} x cos (x) d x = 0

\int_{0}^{2 π} i x sin (x) d x = - 2 iπ

= 0 - (- 2 iπ)

Simplify

= 2 iπ

What is the integral from 0 to 2pi of xe^{-ix} ?
The integral from 0 to 2pi of xe^{-ix} is 2ipi