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

\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

\int sin^{3} (y) cos^{6} (y) d y

d er i v a t i v e b \cdot ln (\frac{x}{a})

\frac{\partial}{\partial y} (\frac{x + 5}{y + 1})

\int e^{x} x^{- 2} d x

d er i v a t i v e \frac{( ( x + 1 ) ( x + 2 ) )}{x}

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