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

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

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

\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) ∣ + C

Solution steps

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

Rewrite using trig identities

= \int \frac{1 + tan ^{2} ( 3 x )}{tan ( 3 x )} d x

Expand

\frac{1 + tan ^{2} ( 3 x )}{tan ( 3 x )} : \frac{1}{tan ( 3 x )} + tan (3 x)

= \int \frac{1}{tan ( 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{1}{tan ( 3 x )} d x + \int tan (3 x) d x

\int \frac{1}{tan ( 3 x )} d x = \frac{1}{3} ln ∣ sin (3 x) ∣

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

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

Add a constant to the solution

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

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