What is the integral of (4-sec^2(3x))/(tan(3x)) ?

The integral of (4-sec^2(3x))/(tan(3x)) is ln|sin(3x)|+1/3 ln|cos(3x)|+C

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integral of (4-sec^2(3x))/(tan(3x))

\int \frac{4 - sec ^{2} ( 3 x )}{tan ( 3 x )} d x

ln ∣ sin (3 x) ∣ + \frac{1}{3} ln ∣ cos (3 x) ∣ + C

Solution steps

\int \frac{4 - sec ^{2} ( 3 x )}{tan ( 3 x )} d x

Expand

\frac{4 - sec ^{2} ( 3 x )}{tan ( 3 x )} : \frac{4}{tan ( 3 x )} - \frac{sec ^{2} ( 3 x )}{tan ( 3 x )}

= \int \frac{4}{tan ( 3 x )} - \frac{sec ^{2} ( 3 x )}{tan ( 3 x )} 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{4}{tan ( 3 x )} d x - \int \frac{sec ^{2} ( 3 x )}{tan ( 3 x )} d x

\int \frac{4}{tan ( 3 x )} d x = \frac{4}{3} ln ∣ sin (3 x) ∣

\int \frac{sec ^{2} ( 3 x )}{tan ( 3 x )} d x = \frac{1}{3} ln ∣ sin (3 x) ∣ - \frac{1}{3} ln ∣ cos (3 x) ∣

= \frac{4}{3} ln ∣ sin (3 x) ∣ - (\frac{1}{3} ln ∣ sin (3 x) ∣ - \frac{1}{3} ln ∣ cos (3 x) ∣)

Simplify

\frac{4}{3} ln ∣ sin (3 x) ∣ - (\frac{1}{3} ln ∣ sin (3 x) ∣ - \frac{1}{3} ln ∣ cos (3 x) ∣) : ln ∣ sin (3 x) ∣ + \frac{1}{3} ln ∣ cos (3 x) ∣

= ln ∣ sin (3 x) ∣ + \frac{1}{3} ln ∣ cos (3 x) ∣

Add a constant to the solution

= ln ∣ sin (3 x) ∣ + \frac{1}{3} ln ∣ cos (3 x) ∣ + C

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What is the integral of (4-sec^2(3x))/(tan(3x)) ?
The integral of (4-sec^2(3x))/(tan(3x)) is ln|sin(3x)|+1/3 ln|cos(3x)|+C