What is the integral of (cos(x))/(4-sin(x)) ?

The integral of (cos(x))/(4-sin(x)) is -ln|4-sin(x)|+C

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integral of (cos(x))/(4-sin(x))

\int \frac{cos ( x )}{4 - sin ( x )} d x

- ln ∣ 4 - sin (x) ∣ + C

Solution steps

\int \frac{cos ( x )}{4 - 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 = 4 - sin (x)

= - ln ∣ 4 - sin (x) ∣

Add a constant to the solution

= - ln ∣ 4 - sin (x) ∣ + C

\int_{1}^{4} x f (x) d x

\int \frac{x ^{2} + 4}{x ^{2}} d x

\int x arctan (x^{2} - 1) d x

\int \frac{x - 1}{x + 1} d x

e^{- x}

What is the integral of (cos(x))/(4-sin(x)) ?
The integral of (cos(x))/(4-sin(x)) is -ln|4-sin(x)|+C