What is the integral of (sin(x))/((2+3cos(x))^2) ?

The integral of (sin(x))/((2+3cos(x))^2) is 1/(3(2+3cos(x)))+C

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integral of (sin(x))/((2+3cos(x))^2)

\int \frac{sin ( x )}{( 2 + 3 cos ( x ) ) ^{2}} d x

\frac{1}{3 ( 2 + 3 cos ( x ) )} + C

Solution steps

\int \frac{sin ( x )}{( 2 + 3 cos ( x ) ) ^{2}} d x

Apply u-substitution

= \int - \frac{1}{( 2 + 3 u ) ^{2}} d u

Take the constant out:

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

= - \int \frac{1}{( 2 + 3 u ) ^{2}} d u

Apply u-substitution

= - \int \frac{1}{3 v ^{2}} d v

Take the constant out:

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

= - \frac{1}{3} \cdot \int \frac{1}{v ^{2}} d v

Apply the Power Rule

= - \frac{1}{3} (- \frac{1}{v})

Substitute back

= - \frac{1}{3} (- \frac{1}{2 + 3 cos ( x )})

Simplify

- \frac{1}{3} (- \frac{1}{2 + 3 cos ( x )}) : \frac{1}{3 ( 2 + 3 cos ( x ) )}

= \frac{1}{3 ( 2 + 3 cos ( x ) )}

Add a constant to the solution

= \frac{1}{3 ( 2 + 3 cos ( x ) )} + C

What is the integral of (sin(x))/((2+3cos(x))^2) ?
The integral of (sin(x))/((2+3cos(x))^2) is 1/(3(2+3cos(x)))+C