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

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

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

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

2 (2 + sin (x) - 2 ln ∣ 2 + sin (x) ∣) + C

Solution steps

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

Rewrite using trig identities

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

Take the constant out:

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

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

Apply u-substitution

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

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

Apply u-substitution

: \int \frac{v - 2}{v} d v

= 2 \cdot \int \frac{v - 2}{v} d v

Expand

\frac{v - 2}{v} : 1 - \frac{2}{v}

= 2 \cdot \int 1 - \frac{2}{v} d v

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= 2 (\int 1 d v - \int \frac{2}{v} d v)

\int 1 d v = v

\int \frac{2}{v} d v = 2 ln ∣ v ∣

= 2 (v - 2 ln ∣ v ∣)

Substitute back

= 2 (2 + sin (x) - 2 ln ∣ 2 + sin (x) ∣)

Add a constant to the solution

= 2 (2 + sin (x) - 2 ln ∣ 2 + sin (x) ∣) + C

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