What is the integral from 0 to 1 of xsin(pinx) ?

The integral from 0 to 1 of xsin(pinx) is -((-1)^n)/(pin)

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral from 0 to 1 of xsin(pinx)

\int_{0}^{1} x sin (πn x) d x

- \frac{( - 1 ) ^{n}}{πn}

Solution steps

\int_{0}^{1} x sin (πn x) d x

Apply Integration By Parts

: [- \frac{1}{πn} x cos (πn x) - \int - \frac{1}{πn} cos (πn x) d x]_{0}^{1}

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

Simplify

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

Compute the boundaries:

- (- 1)^{n} \frac{1}{πn}

= - (- 1)^{n} \frac{1}{πn}

Simplify

= - \frac{( - 1 ) ^{n}}{πn}

\int_{3}^{\infty} (x ln (x)) d x

\int_{0}^{1} e^{3 x^{2}} d x

\int_{0}^{2} r^{2} d r

\int_{0}^{3} \frac{3 x}{1 + x} d x

\int_{0}^{x + y} e^{x} (y + 2 z) d z

What is the integral from 0 to 1 of xsin(pinx) ?
The integral from 0 to 1 of xsin(pinx) is -((-1)^n)/(pin)