What is the integral of (cos(x))/(8-sin(x)) ?

The integral of (cos(x))/(8-sin(x)) is -ln|8-sin(x)|+C

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integral of (cos(x))/(8-sin(x))

\int \frac{cos ( x )}{8 - sin ( x )} d x

- ln ∣ 8 - sin (x) ∣ + C

Solution steps

\int \frac{cos ( x )}{8 - sin ( x )} d x

Apply u-substitution

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

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

Take the constant out:

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

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

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= - ln ∣ u ∣

Substitute back

u = 8 - sin (x)

= - ln ∣ 8 - sin (x) ∣

Add a constant to the solution

= - ln ∣ 8 - sin (x) ∣ + C

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What is the integral of (cos(x))/(8-sin(x)) ?
The integral of (cos(x))/(8-sin(x)) is -ln|8-sin(x)|+C