What is the integral of sin(mx) ?

The integral of sin(mx) is -1/m cos(mx)+C

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integral of sin(mx)

\int sin (m x) d x

- \frac{1}{m} cos (m x) + C

Solution steps

\int sin (m x) d x

Apply u-substitution

= \int \frac{sin ( u )}{m} d u

Take the constant out:

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

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

Use the common integral:

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

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

Substitute back

u = m x

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

Simplify

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

Add a constant to the solution

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

\frac{d y}{d x} = (\frac{y - x + 1}{y - x - 6})

x \to 0 lim (cos (\frac{7}{x}))

d er i v a t i v e f (x) = - 8 x^{2}

d er i v a t i v e y = 12 (ln (2 x) 3)

l a pl a ce t r an s f or m 7 e^{2 t}