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

The integral from 0 to 2 of x^2sin((npix)/2) is (16(-1)^n-16)/(pi^3n^3)-(8(-1)^n)/(pin)

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

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

\frac{16 ( - 1 ) ^{n} - 16}{π ^{3} n ^{3}} - \frac{8 ( - 1 ) ^{n}}{πn}

Solution steps

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

Apply Integration By Parts

: [- \frac{2}{πn} x^{2} cos (\frac{πn}{2} x) - \int - \frac{4}{πn} x cos (\frac{πn}{2} x) d x]_{0}^{2}

= [- \frac{2}{πn} x^{2} cos (\frac{πn}{2} x) - \int - \frac{4}{πn} x cos (\frac{πn}{2} x) d x]_{0}^{2}

\int - \frac{4}{πn} x cos (\frac{πn}{2} x) d x = - \frac{16}{π ^{3} n ^{3}} (\frac{π}{2} n x sin (\frac{π}{2} n x) + cos (\frac{π}{2} n x))

= [- \frac{2}{πn} x^{2} cos (\frac{πn}{2} x) - (- \frac{16}{π ^{3} n ^{3}} (\frac{π}{2} n x sin (\frac{π}{2} n x) + cos (\frac{π}{2} n x)))]_{0}^{2}

Simplify

= [- \frac{2}{πn} x^{2} cos (\frac{πn}{2} x) + \frac{16}{π ^{3} n ^{3}} (\frac{π}{2} n x sin (\frac{π}{2} n x) + cos (\frac{π}{2} n x))]_{0}^{2}

Compute the boundaries:

(- 1)^{n} \frac{16}{π ^{3} n ^{3}} - \frac{8 ( - 1 ) ^{n}}{πn} - \frac{16}{π ^{3} n ^{3}}

= (- 1)^{n} \frac{16}{π ^{3} n ^{3}} - \frac{8 ( - 1 ) ^{n}}{πn} - \frac{16}{π ^{3} n ^{3}}

Simplify

= \frac{16 ( - 1 ) ^{n} - 16}{π ^{3} n ^{3}} - \frac{8 ( - 1 ) ^{n}}{πn}

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What is the integral from 0 to 2 of x^2sin((npix)/2) ?
The integral from 0 to 2 of x^2sin((npix)/2) is (16(-1)^n-16)/(pi^3n^3)-(8(-1)^n)/(pin)