What is the integral from 0 to pi of xsin^3(x) ?

The integral from 0 to pi of xsin^3(x) is (2pi)/3

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^3(x)

\int_{0}^{π} x sin^{3} (x) d x

\frac{2 π}{3}

Decimal

2.09439 \dots

Solution steps

\int_{0}^{π} x sin^{3} (x) d x

Apply Integration By Parts

= [x (- cos (x) + \frac{1}{3} cos^{3} (x)) - \int - cos (x) + \frac{1}{3} cos^{3} (x) d x]_{0}^{π}

\int - cos (x) + \frac{1}{3} cos^{3} (x) d x = - \frac{2}{3} sin (x) - \frac{1}{9} sin^{3} (x)

= [x (- cos (x) + \frac{1}{3} cos^{3} (x)) - (- \frac{2}{3} sin (x) - \frac{1}{9} sin^{3} (x))]_{0}^{π}

Simplify

[x (- cos (x) + \frac{1}{3} cos^{3} (x)) - (- \frac{2}{3} sin (x) - \frac{1}{9} sin^{3} (x))]_{0}^{π} : [\frac{1}{9} (- 9 x cos (x) + 3 x cos^{3} (x) + 6 sin (x) + sin^{3} (x))]_{0}^{π}

= [\frac{1}{9} (- 9 x cos (x) + 3 x cos^{3} (x) + 6 sin (x) + sin^{3} (x))]_{0}^{π}

Compute the boundaries:

\frac{2 π}{3}

= \frac{2 π}{3}

What is the integral from 0 to pi of xsin^3(x) ?
The integral from 0 to pi of xsin^3(x) is (2pi)/3