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

The integral of (3sin(x)+2)/(2sin(x)+3) is 3/2 x-sqrt(5)arctan(1/(sqrt(5))(3tan(x/2)+2))+C

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

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

\frac{3}{2} x - 5 arctan (\frac{1}{5} (3 tan (\frac{x}{2}) + 2)) + C

Solution steps

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

Long division

\frac{3 sin ( x ) + 2}{2 sin ( x ) + 3} : \frac{3}{2} - \frac{5}{2 ( 2 sin ( x ) + 3 )}

= \int \frac{3}{2} - \frac{5}{2 ( 2 sin ( x ) + 3 )} d x

Apply the Sum Rule:

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

= \int \frac{3}{2} d x - \int \frac{5}{2 ( 2 sin ( x ) + 3 )} d x

\int \frac{3}{2} d x = \frac{3}{2} x

\int \frac{5}{2 ( 2 sin ( x ) + 3 )} d x = 5 arctan (\frac{1}{5} (3 tan (\frac{x}{2}) + 2))

= \frac{3}{2} x - 5 arctan (\frac{1}{5} (3 tan (\frac{x}{2}) + 2))

Add a constant to the solution

= \frac{3}{2} x - 5 arctan (\frac{1}{5} (3 tan (\frac{x}{2}) + 2)) + C

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What is the integral of (3sin(x)+2)/(2sin(x)+3) ?
The integral of (3sin(x)+2)/(2sin(x)+3) is 3/2 x-sqrt(5)arctan(1/(sqrt(5))(3tan(x/2)+2))+C