What is the integral of (4sin(3y)-6)/(cos(3y)) ?

The integral of (4sin(3y)-6)/(cos(3y)) is -4/3 ln|cos(3y)|-2ln|tan(3y)+sec(3y)|+C

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integral of (4sin(3y)-6)/(cos(3y))

\int \frac{4 sin ( 3 y ) - 6}{cos ( 3 y )} d y

- \frac{4}{3} ln ∣ cos (3 y) ∣ - 2 ln ∣ tan (3 y) + sec (3 y) ∣ + C

Solution steps

\int \frac{4 sin ( 3 y ) - 6}{cos ( 3 y )} d y

Expand

\frac{4 sin ( 3 y ) - 6}{cos ( 3 y )} : \frac{4 sin ( 3 y )}{cos ( 3 y )} - \frac{6}{cos ( 3 y )}

= \int \frac{4 sin ( 3 y )}{cos ( 3 y )} - \frac{6}{cos ( 3 y )} d y

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{4 sin ( 3 y )}{cos ( 3 y )} d y - \int \frac{6}{cos ( 3 y )} d y

\int \frac{4 sin ( 3 y )}{cos ( 3 y )} d y = - \frac{4}{3} ln ∣ cos (3 y) ∣

\int \frac{6}{cos ( 3 y )} d y = 2 ln ∣ tan (3 y) + sec (3 y) ∣

= - \frac{4}{3} ln ∣ cos (3 y) ∣ - 2 ln ∣ tan (3 y) + sec (3 y) ∣

Add a constant to the solution

= - \frac{4}{3} ln ∣ cos (3 y) ∣ - 2 ln ∣ tan (3 y) + sec (3 y) ∣ + C

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What is the integral of (4sin(3y)-6)/(cos(3y)) ?
The integral of (4sin(3y)-6)/(cos(3y)) is -4/3 ln|cos(3y)|-2ln|tan(3y)+sec(3y)|+C