What is the integral of (cos(x))/(3-sin(x)) ?

The integral of (cos(x))/(3-sin(x)) is -ln|3-sin(x)|+C

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integral of (cos(x))/(3-sin(x))

\int \frac{cos ( x )}{3 - sin ( x )} d x

- ln ∣ 3 - sin (x) ∣ + C

Solution steps

\int \frac{cos ( x )}{3 - 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 = 3 - sin (x)

= - ln ∣ 3 - sin (x) ∣

Add a constant to the solution

= - ln ∣ 3 - sin (x) ∣ + C

\int \frac{1}{e ^{2 y} - 1} d y

\int_{0}^{\frac{π}{4}} x^{2} sin (2 x) d x

\frac{d}{d x} (cos (x) + ln (cos (x)) + x sin (x))

\int \frac{x ^{3}}{1 + x ^{3}} d x

x \to 1 lim (x^{2} + 2 x - 2)

What is the integral of (cos(x))/(3-sin(x)) ?
The integral of (cos(x))/(3-sin(x)) is -ln|3-sin(x)|+C