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

The integral of t^2sin(2t) is -1/2 t^2cos(2t)+1/4 (2tsin(2t)+cos(2t))+C

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integral of t^2sin(2t)

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

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

Solution steps

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

Apply Integration By Parts

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

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

\int - t cos (2 t) d t = - \frac{1}{4} (2 t sin (2 t) + cos (2 t))

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

Simplify

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

Add a constant to the solution

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

f (x) = x^{\frac{1}{4}}

x \to + \frac{π}{2} lim (tan (x))

2 + 3 x

y^{^{'}} - 2 y = t^{2} e^{2 t}

y^{^{''}} + y = 7 sin (2 t) + 5 t cos (2 t)

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