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

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

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integral of sin(-5x)

\int sin (- 5 x) d x

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

Solution steps

\int sin (- 5 x) d x

Apply u-substitution

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

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

Use the common integral:

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

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

Substitute back

u = - 5 x

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

Simplify

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

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

Add a constant to the solution

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

\int (e^{5 x}) d x

\int x + 3 - 3 d x

x + y y^{^{'}} = 0

y = x^{3} + 7 x^{2} - 4

\frac{d y}{d t} = 4 t - 0.1 y t