What is the integral from 0 to 3 of xsin((npix)/3) ?

The integral from 0 to 3 of xsin((npix)/3) is -(9(-1)^n)/(pin)

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integral from 0 to 3 of xsin((npix)/3)

\int_{0}^{3} x sin (\frac{nπ x}{3}) d x

- \frac{9 ( - 1 ) ^{n}}{πn}

Solution steps

\int_{0}^{3} x sin (\frac{nπ x}{3}) d x

Apply u-substitution

: \int_{0}^{nπ 3} \frac{u sin ( \frac{u}{3} )}{π ^{2} n ^{2}} d u

= \int_{0}^{nπ 3} \frac{u sin ( \frac{u}{3} )}{π ^{2} n ^{2}} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{π ^{2} n ^{2}} \cdot \int_{0}^{nπ 3} u sin (\frac{u}{3}) d u

Apply Integration By Parts

: [- 3 u cos (\frac{u}{3}) - \int - 3 cos (\frac{u}{3}) d u]_{0}^{3 πn}

= \frac{1}{π ^{2} n ^{2}} [- 3 u cos (\frac{u}{3}) - \int - 3 cos (\frac{u}{3}) d u]_{0}^{3 πn}

\int - 3 cos (\frac{u}{3}) d u = - 9 sin (\frac{u}{3})

= \frac{1}{π ^{2} n ^{2}} [- 3 u cos (\frac{u}{3}) - (- 9 sin (\frac{u}{3}))]_{0}^{3 πn}

Simplify

= \frac{1}{π ^{2} n ^{2}} [- 3 u cos (\frac{u}{3}) + 9 sin (\frac{u}{3})]_{0}^{3 πn}

Compute the boundaries:

- 9 π (- 1)^{n} n

= \frac{1}{π ^{2} n ^{2}} (- 9 π (- 1)^{n} n)

Simplify

= - \frac{9 ( - 1 ) ^{n}}{πn}

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What is the integral from 0 to 3 of xsin((npix)/3) ?
The integral from 0 to 3 of xsin((npix)/3) is -(9(-1)^n)/(pin)