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

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

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integral of (cos^2(3x))/(1-sin(3x))

\int \frac{cos ^{2} ( 3 x )}{1 - sin ( 3 x )} d x

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

Solution steps

\int \frac{cos ^{2} ( 3 x )}{1 - sin ( 3 x )} d x

Rewrite using trig identities

= \int sin (3 x) + 1 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 sin (3 x) d x + \int 1 d x

\int sin (3 x) d x = - \frac{1}{3} cos (3 x)

\int 1 d x = x

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

Add a constant to the solution

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

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