What is the integral of 5sin(-2x) ?

The integral of 5sin(-2x) is 5/2 cos(2x)+C

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integral of 5sin(-2x)

\int 5 sin (- 2 x) d x

\frac{5}{2} cos (2 x) + C

Solution steps

\int 5 sin (- 2 x) d x

Take the constant out:

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

= 5 \cdot \int sin (- 2 x) d x

Apply u-substitution

= 5 \cdot \int - \frac{1}{2} sin (u) d u

Take the constant out:

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

= 5 (- \frac{1}{2} \cdot \int sin (u) d u)

Use the common integral:

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

= 5 (- \frac{1}{2} (- cos (u)))

Substitute back

u = - 2 x

= 5 (- \frac{1}{2} (- cos (- 2 x)))

Simplify

5 (- \frac{1}{2} (- cos (- 2 x))) : \frac{5}{2} cos (2 x)

= \frac{5}{2} cos (2 x)

Add a constant to the solution

= \frac{5}{2} cos (2 x) + C