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

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

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

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

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

Solution steps

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

Apply Integration By Parts

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

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

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

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

Compute the boundaries:

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

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

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\int_{\frac{π}{3}}^{\frac{π}{2}} 1 d x

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