What is the integral of 2tsin(2t) ?

The integral of 2tsin(2t) is -tcos(2t)+1/2 sin(2t)+C

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integral of 2tsin(2t)

\int 2 t sin (2 t) d t

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

Solution steps

\int 2 t sin (2 t) d t

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 2 \cdot \int t sin (2 t) d t

Apply Integration By Parts

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

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

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

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

Simplify

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

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

Add a constant to the solution

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

\int 2

\int (arctan (\frac{1}{x})) d x

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