What is the integral of x^3sin(nx) ?

The integral of x^3sin(nx) is -1/n x^3cos(nx)+3/(n^4)(n^2x^2sin(nx)-2(-nxcos(nx)+sin(nx)))+C

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integral of x^3sin(nx)

\int x^{3} sin (n x) d x

- \frac{1}{n} x^{3} cos (n x) + \frac{3}{n ^{4}} (n^{2} x^{2} sin (n x) - 2 (- n x cos (n x) + sin (n x))) + C

Solution steps

\int x^{3} sin (n x) d x

Apply Integration By Parts

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

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

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

Simplify

= - \frac{1}{n} x^{3} cos (n x) + \frac{3}{n ^{4}} (n^{2} x^{2} sin (n x) - 2 (- n x cos (n x) + sin (n x)))

Add a constant to the solution

= - \frac{1}{n} x^{3} cos (n x) + \frac{3}{n ^{4}} (n^{2} x^{2} sin (n x) - 2 (- n x cos (n x) + sin (n x))) + C

x \frac{d ^{2} y}{d x ^{2}} + 2 \frac{d y}{d x} = 0

\int_{3}^{5} \frac{x}{x ^{2} + 6 x + 13} d x

x \to 0 lim (\frac{- 2 x ( x + 2 )}{4 x ^{2} - 3 x - 7})

\frac{d}{d x} (3 x^{3} + 2)

f (x) = - 4 - 5 x^{2}, (- 5, - 129)

What is the integral of x^3sin(nx) ?
The integral of x^3sin(nx) is -1/n x^3cos(nx)+3/(n^4)(n^2x^2sin(nx)-2(-nxcos(nx)+sin(nx)))+C