What is the integral of (sin(kt))/(a+cos(kt)) ?

The integral of (sin(kt))/(a+cos(kt)) is -1/k ln|a+cos(kt)|+C

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integral of (sin(kt))/(a+cos(kt))

\int \frac{sin ( k t )}{a + cos ( k t )} d t

- \frac{1}{k} ln ∣ a + cos (k t) ∣ + C

Solution steps

\int \frac{sin ( k t )}{a + cos ( k t )} d t

Apply u-substitution

: \int - \frac{1}{k u} d u

= \int - \frac{1}{k u} d u

Take the constant out:

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

= - \frac{1}{k} \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= - \frac{1}{k} ln ∣ u ∣

Substitute back

u = a + cos (k t)

= - \frac{1}{k} ln ∣ a + cos (k t) ∣

Add a constant to the solution

= - \frac{1}{k} ln ∣ a + cos (k t) ∣ + C

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What is the integral of (sin(kt))/(a+cos(kt)) ?
The integral of (sin(kt))/(a+cos(kt)) is -1/k ln|a+cos(kt)|+C