What is the integral from 0 to pi of xsin(kx) ?

The integral from 0 to pi of xsin(kx) is (sin(pik)-pikcos(pik))/(k^2)

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

integral from 0 to pi of xsin(kx)

\int_{0}^{π} x sin (k x) d x

\frac{sin ( πk ) - πk cos ( πk )}{k ^{2}}

Solution steps

\int_{0}^{π} x sin (k x) d x

Apply Integration By Parts

= [\frac{1}{k} (- x cos (k x) - k \cdot \int - \frac{1}{k} cos (k x) d x)]_{0}^{π}

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

= [\frac{1}{k} (- x cos (k x) - k (- \frac{1}{k ^{2}} sin (k x)))]_{0}^{π}

Simplify

[\frac{1}{k} (- x cos (k x) - k (- \frac{1}{k ^{2}} sin (k x)))]_{0}^{π} : [\frac{1}{k} (- x cos (k x) + \frac{1}{k} sin (k x))]_{0}^{π}

= [\frac{1}{k} (- x cos (k x) + \frac{1}{k} sin (k x))]_{0}^{π}

Compute the boundaries:

\frac{1}{k} (\frac{1}{k} sin (πk) - π cos (πk))

= \frac{1}{k} (\frac{1}{k} sin (πk) - π cos (πk))

Simplify

= \frac{sin ( πk ) - πk cos ( πk )}{k ^{2}}

What is the integral from 0 to pi of xsin(kx) ?
The integral from 0 to pi of xsin(kx) is (sin(pik)-pikcos(pik))/(k^2)