What is the integral from 0 to 3 of x^2cos((npix)/3) ?

The integral from 0 to 3 of x^2cos((npix)/3) is (54(-1)^n)/(pi^2n^2)

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integral from 0 to 3 of x^2cos((npix)/3)

\int_{0}^{3} x^{2} cos (\frac{nπ x}{3}) d x

\frac{54 ( - 1 ) ^{n}}{π ^{2} n ^{2}}

Solution steps

\int_{0}^{3} x^{2} cos (\frac{nπ x}{3}) d x

Apply Integration By Parts

: [\frac{3}{πn} x^{2} sin (\frac{πn x}{3}) - \int \frac{6}{πn} x sin (\frac{πn x}{3}) d x]_{0}^{3}

= [\frac{3}{πn} x^{2} sin (\frac{πn x}{3}) - \int \frac{6}{πn} x sin (\frac{πn x}{3}) d x]_{0}^{3}

\int \frac{6}{πn} x sin (\frac{πn x}{3}) d x = \frac{6}{π ^{3} n ^{3}} (- 3 πn x cos (\frac{πn x}{3}) + 9 sin (\frac{πn x}{3}))

= [\frac{3}{πn} x^{2} sin (\frac{πn x}{3}) - \frac{6}{π ^{3} n ^{3}} (- 3 πn x cos (\frac{πn x}{3}) + 9 sin (\frac{πn x}{3}))]_{0}^{3}

Compute the boundaries:

\frac{54 ( - 1 ) ^{n}}{π ^{2} n ^{2}}

= \frac{54 ( - 1 ) ^{n}}{π ^{2} n ^{2}}

\int_{0}^{2} 16 x - x^{5} d x

\int_{1}^{2} \frac{3}{7} x^{2} x^{2} d x

\int_{2}^{4} 3 + x^{2} d x

\int_{0}^{π} sin (2 x) cos (n x) d x

\int_{2}^{6} 6 x^{2} ln (3 x) d x

What is the integral from 0 to 3 of x^2cos((npix)/3) ?
The integral from 0 to 3 of x^2cos((npix)/3) is (54(-1)^n)/(pi^2n^2)