What is the integral of t^3sin(nt) ?

The integral of t^3sin(nt) is -1/n t^3cos(nt)+3/(n^4)(n^2t^2sin(nt)-2(-ntcos(nt)+sin(nt)))+C

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integral of t^3sin(nt)

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

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

Solution steps

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

Apply Integration By Parts

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

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

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

Simplify

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

Add a constant to the solution

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

What is the integral of t^3sin(nt) ?
The integral of t^3sin(nt) is -1/n t^3cos(nt)+3/(n^4)(n^2t^2sin(nt)-2(-ntcos(nt)+sin(nt)))+C