What is the integral of t^2sin(nt) ?

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

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integral of t^2sin(nt)

\int t^{2} sin (n t) d t

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

Solution steps

\int t^{2} sin (n t) d t

Apply Integration By Parts

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

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

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

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

Simplify

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

Add a constant to the solution

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

x \to e + lim ((ln (x))^{\frac{1}{x - e}})

y^{^{''}} + 9 y = e^{t} (cos (3 t) + sin (3 t))

x \to 9 lim (7 x + ∣ x - 9 ∣)

f (x) = \frac{x ^{3}}{4} - 3 x

\frac{\partial}{\partial x} (y^{3} cos (3 x) \cdot 3)

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