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

The integral of t^3sin(t) is -t^3cos(t)+3(t^2sin(t)-2(-tcos(t)+sin(t)))+C

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integral of t^3sin(t)

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

- t^{3} cos (t) + 3 (t^{2} sin (t) - 2 (- t cos (t) + sin (t))) + C

Solution steps

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

Apply Integration By Parts

: - t^{3} cos (t) - \int - 3 t^{2} cos (t) d t

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

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

= - t^{3} cos (t) - (- 3 (t^{2} sin (t) - 2 (- t cos (t) + sin (t))))

Simplify

= - t^{3} cos (t) + 3 (t^{2} sin (t) - 2 (- t cos (t) + sin (t)))

Add a constant to the solution

= - t^{3} cos (t) + 3 (t^{2} sin (t) - 2 (- t cos (t) + sin (t))) + C

\frac{d y}{d x} + 2 y = e^{2 x} y^{- 2}

y^{^{'}} + \frac{1}{x} y = 6 x^{2}

\frac{\partial}{\partial y} (x^{2} - 4 x y + 3 y^{2})

\frac{d}{d x} ((4 x)^{x})

\int 10 x^{4} - 2 sec^{2} (x) d x

What is the integral of t^3sin(t) ?
The integral of t^3sin(t) is -t^3cos(t)+3(t^2sin(t)-2(-tcos(t)+sin(t)))+C