What is the integral from 0 to 3 of x^2sin((npix)/3) ?

The integral from 0 to 3 of x^2sin((npix)/3) is -(27(-1)^n)/(pin)+(54(-1)^n-54)/(pi^3n^3)

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integral from 0 to 3 of x^2sin((npix)/3)

\int_{0}^{3} x^{2} sin (\frac{nπ x}{3}) d x

- \frac{27 ( - 1 ) ^{n}}{πn} + \frac{54 ( - 1 ) ^{n} - 54}{π ^{3} n ^{3}}

Solution steps

\int_{0}^{3} x^{2} sin (\frac{nπ x}{3}) d x

Apply Integration By Parts

: [- \frac{3}{πn} x^{2} cos (\frac{πn x}{3}) - \int - \frac{6}{πn} x cos (\frac{πn x}{3}) d x]_{0}^{3}

= [- \frac{3}{πn} x^{2} cos (\frac{πn x}{3}) - \int - \frac{6}{πn} x cos (\frac{πn x}{3}) d x]_{0}^{3}

\int - \frac{6}{πn} x cos (\frac{πn x}{3}) d x = - \frac{6}{π ^{3} n ^{3}} (3 πn x sin (\frac{πn x}{3}) + 9 cos (\frac{πn x}{3}))

= [- \frac{3}{πn} x^{2} cos (\frac{πn x}{3}) - (- \frac{6}{π ^{3} n ^{3}} (3 πn x sin (\frac{πn x}{3}) + 9 cos (\frac{πn x}{3})))]_{0}^{3}

Simplify

= [- \frac{3}{πn} x^{2} cos (\frac{πn x}{3}) + \frac{6}{π ^{3} n ^{3}} (3 πn x sin (\frac{πn x}{3}) + 9 cos (\frac{πn x}{3}))]_{0}^{3}

Compute the boundaries:

- \frac{27 ( - 1 ) ^{n}}{πn} + \frac{54 ( - 1 ) ^{n}}{π ^{3} n ^{3}} - \frac{54}{π ^{3} n ^{3}}

= - \frac{27 ( - 1 ) ^{n}}{πn} + \frac{54 ( - 1 ) ^{n}}{π ^{3} n ^{3}} - \frac{54}{π ^{3} n ^{3}}

Simplify

= - \frac{27 ( - 1 ) ^{n}}{πn} + \frac{54 ( - 1 ) ^{n} - 54}{π ^{3} n ^{3}}

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What is the integral from 0 to 3 of x^2sin((npix)/3) ?
The integral from 0 to 3 of x^2sin((npix)/3) is -(27(-1)^n)/(pin)+(54(-1)^n-54)/(pi^3n^3)