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

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

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

\int_{0}^{π} x^{3} sin (n x) d x

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

Solution steps

\int_{0}^{π} x^{3} sin (n x) d x

Apply Integration By Parts

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

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

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

= [\frac{1}{n} (- x^{3} cos (n x) - n (- \frac{3}{n ^{4}} (n^{2} x^{2} sin (n x) - 2 (- n x cos (n x) + sin (n x)))))]_{0}^{π}

Simplify

[\frac{1}{n} (- x^{3} cos (n x) - n (- \frac{3}{n ^{4}} (n^{2} x^{2} sin (n x) - 2 (- n x cos (n x) + sin (n x)))))]_{0}^{π} : [\frac{1}{n} (- x^{3} cos (n x) + \frac{3}{n ^{3}} (n^{2} x^{2} sin (n x) - 2 (- n x cos (n x) + sin (n x))))]_{0}^{π}

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

Compute the boundaries:

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

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

Simplify

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

\int_{0}^{0.4} ln (1 + x^{2}) d x

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\int_{\frac{π}{2}}^{0} sin (x) d x

\int_{0}^{\frac{π}{2}} cos (2 u) d u

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