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

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

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

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

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

Solution steps

\int_{0}^{l} 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]_{0}^{l}

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

\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}))]_{0}^{l}

Compute the boundaries:

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

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

\int_{0}^{\frac{π}{2}} cos^{2} (s) in x d x

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\int_{a}^{b} \frac{1}{x} d x

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\int_{100}^{v} 1 d v

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