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

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

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integral of-sin(2t)

\int - sin (2 t) d t

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

Solution steps

\int - sin (2 t) d t

Take the constant out:

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

= - \int sin (2 t) d t

Apply u-substitution

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

Take the constant out:

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

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

Use the common integral:

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

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

Substitute back

u = 2 t

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

Simplify

= \frac{1}{2} cos (2 t)

Add a constant to the solution

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