What is the integral from 0 to pi/n of xsin(nx) ?

The integral from 0 to pi/n of xsin(nx) is pi/(n^2)

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integral from 0 to pi/n of xsin(nx)

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

\frac{π}{n ^{2}}

Solution steps

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

Apply Integration By Parts

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

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

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

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

Simplify

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

Compute the boundaries:

\frac{π}{n ^{2}}

= \frac{π}{n ^{2}}

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\int_{0}^{1} θ x^{θ - 1} d x

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