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

The integral from-pi to pi of x^3sin(nx) is (12pi(-1)^n}{n^3}-\frac{2pi^3(-1)^n)/n

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

\int_{- π}^{π} x^{3} sin (n x) d x

\frac{12 π ( - 1 ) ^{n}}{n ^{3}} - \frac{2 π ^{3} ( - 1 ) ^{n}}{n}

Solution steps

\int_{- π}^{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)]_{- π}^{π}

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

Compute the boundaries:

\frac{12 π ( - 1 ) ^{n}}{n ^{3}} - 2 (- 1)^{n} \frac{π ^{3}}{n}

= \frac{12 π ( - 1 ) ^{n}}{n ^{3}} - 2 (- 1)^{n} \frac{π ^{3}}{n}

Simplify

= \frac{12 π ( - 1 ) ^{n}}{n ^{3}} - \frac{2 π ^{3} ( - 1 ) ^{n}}{n}

7 - x - x^{2}

\frac{d}{d x} (\frac{x}{5 + x ^{2}})

y^{^{''}} + 9 y = 4 tan (3 x)

y = ln^{2} (x + 3)

\frac{\partial}{\partial x} (e^{x^{2} y})

What is the integral from-pi to pi of x^3sin(nx) ?
The integral from-pi to pi of x^3sin(nx) is (12pi(-1)^n}{n^3}-\frac{2pi^3(-1)^n)/n