What is the integral from 0 to pi of x^3cos(nx) ?

The integral from 0 to pi of x^3cos(nx) is (-3(2(-1)^n-pi^2(-1)^nn^2)+6)/(n^4)

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integral from 0 to pi of x^3cos(nx)

\int_{0}^{π} x^{3} cos (n x) d x

\frac{- 3 ( 2 ( - 1 ) ^{n} - π ^{2} ( - 1 ) ^{n} n ^{2} ) + 6}{n ^{4}}

Solution steps

\int_{0}^{π} x^{3} cos (n x) d x

Apply Integration By Parts

: [\frac{1}{n} (x^{3} sin (n x) - n \cdot \int \frac{3}{n} x^{2} sin (n x) d x)]_{0}^{π}

= [\frac{1}{n} (x^{3} sin (n x) - n \cdot \int \frac{3}{n} x^{2} sin (n x) d x)]_{0}^{π}

\int \frac{3}{n} x^{2} sin (n x) d x = \frac{3}{n} (- \frac{1}{n} x^{2} cos (n x) + \frac{2}{n ^{3}} (n x sin (n x) + cos (n x)))

= [\frac{1}{n} (x^{3} sin (n x) - n \frac{3}{n} (- \frac{1}{n} x^{2} cos (n x) + \frac{2}{n ^{3}} (n x sin (n x) + cos (n x))))]_{0}^{π}

Simplify

[\frac{1}{n} (x^{3} sin (n x) - n \frac{3}{n} (- \frac{1}{n} x^{2} cos (n x) + \frac{2}{n ^{3}} (n x sin (n x) + cos (n x))))]_{0}^{π} : [\frac{1}{n} (x^{3} sin (n x) - 3 (- \frac{1}{n} x^{2} cos (n x) + \frac{2}{n ^{3}} (n x sin (n x) + cos (n x))))]_{0}^{π}

= [\frac{1}{n} (x^{3} sin (n x) - 3 (- \frac{1}{n} x^{2} cos (n x) + \frac{2}{n ^{3}} (n x sin (n x) + cos (n x))))]_{0}^{π}

Compute the boundaries:

- \frac{3}{n} ((- 1)^{n} \frac{2}{n ^{3}} - (- 1)^{n} \frac{π ^{2}}{n}) + \frac{6}{n ^{4}}

= - \frac{3}{n} ((- 1)^{n} \frac{2}{n ^{3}} - (- 1)^{n} \frac{π ^{2}}{n}) + \frac{6}{n ^{4}}

Simplify

= \frac{- 3 ( 2 ( - 1 ) ^{n} - π ^{2} ( - 1 ) ^{n} n ^{2} ) + 6}{n ^{4}}

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What is the integral from 0 to pi of x^3cos(nx) ?
The integral from 0 to pi of x^3cos(nx) is (-3(2(-1)^n-pi^2(-1)^nn^2)+6)/(n^4)