What is the integral from-1 to 1 of xsin(npix) ?

The integral from-1 to 1 of xsin(npix) is -(2(-1)^n)/(pin)

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

integral from-1 to 1 of xsin(npix)

\int_{- 1}^{1} x sin (nπ x) d x

- \frac{2 ( - 1 ) ^{n}}{πn}

Solution steps

\int_{- 1}^{1} x sin (nπ x) d x

Apply Integration By Parts

= [- \frac{1}{πn} x cos (πn x) - \int - \frac{1}{πn} cos (πn x) d x]_{- 1}^{1}

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

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

Simplify

= [- \frac{1}{πn} x cos (πn x) + \frac{1}{π ^{2} n ^{2}} sin (πn x)]_{- 1}^{1}

Compute the boundaries:

- (- 1)^{n} \frac{2}{πn}

= - (- 1)^{n} \frac{2}{πn}

Simplify

= - \frac{2 ( - 1 ) ^{n}}{πn}

What is the integral from-1 to 1 of xsin(npix) ?
The integral from-1 to 1 of xsin(npix) is -(2(-1)^n)/(pin)