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

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

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

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

- \frac{( - 1 ) ^{n} L ^{3}}{πn} + \frac{2 ( - 1 ) ^{n} L ^{3} - 2 L ^{3}}{π ^{3} n ^{3}}

Solution steps

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

Apply Integration By Parts

= [- L \frac{1}{πn} x^{2} cos (\frac{πn x}{L}) - \int - L \frac{2}{πn} x cos (\frac{πn x}{L}) d x]_{0}^{L}

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

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

Simplify

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

Compute the boundaries:

- \frac{( - 1 ) ^{n} L ^{3}}{πn} + \frac{2 ( - 1 ) ^{n} L ^{3}}{π ^{3} n ^{3}} - \frac{2 L ^{3}}{π ^{3} n ^{3}}

= - \frac{( - 1 ) ^{n} L ^{3}}{πn} + \frac{2 ( - 1 ) ^{n} L ^{3}}{π ^{3} n ^{3}} - \frac{2 L ^{3}}{π ^{3} n ^{3}}

Simplify

= - \frac{( - 1 ) ^{n} L ^{3}}{πn} + \frac{2 ( - 1 ) ^{n} L ^{3} - 2 L ^{3}}{π ^{3} n ^{3}}

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