What is the integral of 4tsin(2t) ?

The integral of 4tsin(2t) is -2tcos(2t)+sin(2t)+C

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integral of 4tsin(2t)

\int 4 t sin (2 t) d t

- 2 t cos (2 t) + sin (2 t) + C

Solution steps

\int 4 t sin (2 t) d t

Take the constant out:

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

= 4 \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

= 4 (- \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)

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

Simplify

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

= - 2 t cos (2 t) + sin (2 t)

Add a constant to the solution

= - 2 t cos (2 t) + sin (2 t) + C

\int sec^{2} (x) e^{4 t a n (x)} d x

\int \frac{x ^{2} - x ^{3}}{2 x} d x

\int \frac{sin ( I n x )}{x} d x

\int \frac{x}{x ^{4} + 4 x ^{2} + 8} d x

\int x^{2} \cdot e^{x^{2}} \cdot x d x

What is the integral of 4tsin(2t) ?
The integral of 4tsin(2t) is -2tcos(2t)+sin(2t)+C