What is the integral of cos(-5x) ?

The integral of cos(-5x) is 1/5 sin(5x)+C

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integral of cos(-5x)

\int cos (- 5 x) d x

\frac{1}{5} sin (5 x) + C

Solution steps

\int cos (- 5 x) d x

Apply u-substitution

: \int - \frac{1}{5} cos (u) d u

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

Take the constant out:

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

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

Use the common integral:

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

= - \frac{1}{5} sin (u)

Substitute back

u = - 5 x

= - \frac{1}{5} sin (- 5 x)

Simplify

- \frac{1}{5} sin (- 5 x) : \frac{1}{5} sin (5 x)

= \frac{1}{5} sin (5 x)

Add a constant to the solution

= \frac{1}{5} sin (5 x) + C

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