What is the integral of 2sin(3t) ?

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

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integral of 2sin(3t)

\int 2 sin (3 t) d t

- \frac{2}{3} cos (3 t) + C

Solution steps

\int 2 sin (3 t) d t

Take the constant out:

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

= 2 \cdot \int sin (3 t) d t

Apply u-substitution

: \int sin (u) \frac{1}{3} d u

= 2 \cdot \int sin (u) \frac{1}{3} d u

Take the constant out:

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

= 2 \cdot \frac{1}{3} \cdot \int sin (u) d u

Use the common integral:

\int sin (u) d u = - cos (u)

= 2 \cdot \frac{1}{3} (- cos (u))

Substitute back

u = 3 t

= 2 \cdot \frac{1}{3} (- cos (3 t))

Simplify

2 \cdot \frac{1}{3} (- cos (3 t)) : - \frac{2}{3} cos (3 t)

= - \frac{2}{3} cos (3 t)

Add a constant to the solution

= - \frac{2}{3} cos (3 t) + C

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x \to 2 lim (\frac{x - 4}{x - 2})

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\frac{d}{d x} (\frac{tan ( x )}{cos ( x ) - 4})