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

The integral from 0 to pi of x*sin(nx) is -(pi(-1)^n)/n

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integral from 0 to pi of x*sin(nx)

\int_{0}^{π} x \cdot sin (n x) d x

- \frac{π ( - 1 ) ^{n}}{n}

Solution steps

\int_{0}^{π} x sin (n x) d x

Apply Integration By Parts

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

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

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

Simplify

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

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

Compute the boundaries:

- (- 1)^{n} \frac{π}{n}

= - (- 1)^{n} \frac{π}{n}

Simplify

= - \frac{π ( - 1 ) ^{n}}{n}

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