What is the integral from 0 to 1 of x*sin(n*pi*x) ?

The integral from 0 to 1 of x*sin(n*pi*x) 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(npi*x)

\int_{0}^{1} x \cdot sin (n \cdot π \cdot x) d x

- \frac{( - 1 ) ^{n}}{πn}

Solution steps

\int_{0}^{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]_{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_{- π}^{0} (π + x) cos (n x) d x

\int_{0}^{zy} (z) d z

\int_{0}^{3} (3 - t) t d t

\int_{1}^{5} 2 x (3 x^{2} + 5) d x

\int_{0}^{1} 50 x (2 x^{2} + 1)^{4} d x

What is the integral from 0 to 1 of x*sin(n*pi*x) ?
The integral from 0 to 1 of x*sin(n*pi*x) is -((-1)^n)/(pin)