What is the integral of (cos(x))/(1+2sin(x)) ?

The integral of (cos(x))/(1+2sin(x)) is 1/2 ln|1+2sin(x)|+C

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integral of (cos(x))/(1+2sin(x))

\int \frac{cos ( x )}{1 + 2 sin ( x )} d x

\frac{1}{2} ln ∣ 1 + 2 sin (x) ∣ + C

Solution steps

\int \frac{cos ( x )}{1 + 2 sin ( x )} d x

Apply u-substitution

: \int \frac{1}{2 u} d u

= \int \frac{1}{2 u} d u

Take the constant out:

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

= \frac{1}{2} \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= \frac{1}{2} ln ∣ u ∣

Substitute back

u = 1 + 2 sin (x)

= \frac{1}{2} ln ∣ 1 + 2 sin (x) ∣

Add a constant to the solution

= \frac{1}{2} ln ∣ 1 + 2 sin (x) ∣ + C

\frac{d}{d x} (e^{l n (x^{2})})

x \to 0 lim (\frac{1}{x ^{3}})

\frac{d}{d x} (\frac{x}{( 1 - x ) ^{2}})

f (x) = \frac{7}{r ^{3}}

\int \frac{cos ( x )}{1 + 7 sin ( x )} d x

What is the integral of (cos(x))/(1+2sin(x)) ?
The integral of (cos(x))/(1+2sin(x)) is 1/2 ln|1+2sin(x)|+C