What is the integral from 0 to 4 of x^2cos((npix)/4) ?

The integral from 0 to 4 of x^2cos((npix)/4) is (128(-1)^n)/(pi^2n^2)

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integral from 0 to 4 of x^2cos((npix)/4)

\int_{0}^{4} x^{2} cos (\frac{nπ x}{4}) d x

\frac{128 ( - 1 ) ^{n}}{π ^{2} n ^{2}}

Solution steps

\int_{0}^{4} x^{2} cos (\frac{nπ x}{4}) d x

Apply Integration By Parts

: [\frac{4}{πn} x^{2} sin (\frac{πn x}{4}) - \int \frac{8}{πn} x sin (\frac{πn x}{4}) d x]_{0}^{4}

= [\frac{4}{πn} x^{2} sin (\frac{πn x}{4}) - \int \frac{8}{πn} x sin (\frac{πn x}{4}) d x]_{0}^{4}

\int \frac{8}{πn} x sin (\frac{πn x}{4}) d x = \frac{8}{π ^{3} n ^{3}} (- 4 πn x cos (\frac{πn x}{4}) + 16 sin (\frac{πn x}{4}))

= [\frac{4}{πn} x^{2} sin (\frac{πn x}{4}) - \frac{8}{π ^{3} n ^{3}} (- 4 πn x cos (\frac{πn x}{4}) + 16 sin (\frac{πn x}{4}))]_{0}^{4}

Compute the boundaries:

\frac{128 ( - 1 ) ^{n}}{π ^{2} n ^{2}}

= \frac{128 ( - 1 ) ^{n}}{π ^{2} n ^{2}}

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What is the integral from 0 to 4 of x^2cos((npix)/4) ?
The integral from 0 to 4 of x^2cos((npix)/4) is (128(-1)^n)/(pi^2n^2)