What is the integral from-2pi to 2pi of xsin(0.5nx) ?

The integral from-2pi to 2pi of xsin(0.5nx) is -(25.13274…(-1)^n)/n

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integral from-2pi to 2pi of xsin(0.5nx)

\int_{- 2 π}^{2 π} x sin (0.5 n x) d x

- \frac{25.13274 \dots ( - 1 ) ^{n}}{n}

Solution steps

\int_{- 2 π}^{2 π} x sin (0.5 n x) d x

Apply Integration By Parts

: [\frac{1}{n} (- 2 x cos (0.5 n x) - n \cdot \int - \frac{2}{n} cos (0.5 n x) d x)]_{- 6.28318 \dots}^{2 π}

= [\frac{1}{n} (- 2 x cos (0.5 n x) - n \cdot \int - \frac{2}{n} cos (0.5 n x) d x)]_{- 6.28318 \dots}^{2 π}

\int - \frac{2}{n} cos (0.5 n x) d x = - \frac{4}{n ^{2}} sin (0.5 n x)

= [\frac{1}{n} (- 2 x cos (0.5 n x) - n (- \frac{4}{n ^{2}} sin (0.5 n x)))]_{- 6.28318 \dots}^{2 π}

Simplify

[\frac{1}{n} (- 2 x cos (0.5 n x) - n (- \frac{4}{n ^{2}} sin (0.5 n x)))]_{- 6.28318 \dots}^{2 π} : [\frac{1}{n} (- 2 x cos (0.5 n x) + \frac{4}{n} sin (0.5 n x))]_{- 6.28318 \dots}^{2 π}

= [\frac{1}{n} (- 2 x cos (0.5 n x) + \frac{4}{n} sin (0.5 n x))]_{- 6.28318 \dots}^{2 π}

Compute the boundaries:

- (- 1)^{n} \frac{25.13274 \dots}{n}

= - (- 1)^{n} \frac{25.13274 \dots}{n}

Simplify

= - \frac{25.13274 \dots ( - 1 ) ^{n}}{n}

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What is the integral from-2pi to 2pi of xsin(0.5nx) ?
The integral from-2pi to 2pi of xsin(0.5nx) is -(25.13274…(-1)^n)/n