What is the integral of (10sec^2(3x))/(tan(3x)-3) ?

The integral of (10sec^2(3x))/(tan(3x)-3) is 10/3 ln|tan(3x)-3|+C

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integral of (10sec^2(3x))/(tan(3x)-3)

\int \frac{10 sec ^{2} ( 3 x )}{tan ( 3 x ) - 3} d x

\frac{10}{3} ln ∣ tan (3 x) - 3 ∣ + C

Solution steps

\int \frac{10 sec ^{2} ( 3 x )}{tan ( 3 x ) - 3} d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 10 \cdot \int \frac{sec ^{2} ( 3 x )}{tan ( 3 x ) - 3} d x

Apply u-substitution

= 10 \cdot \int \frac{1}{3 u} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 10 \cdot \frac{1}{3} \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 10 \cdot \frac{1}{3} ln ∣ u ∣

Substitute back

u = tan (3 x) - 3

= 10 \cdot \frac{1}{3} ln ∣ tan (3 x) - 3 ∣

Simplify

10 \cdot \frac{1}{3} ln ∣ tan (3 x) - 3 ∣ : \frac{10}{3} ln ∣ tan (3 x) - 3 ∣

= \frac{10}{3} ln ∣ tan (3 x) - 3 ∣

Add a constant to the solution

= \frac{10}{3} ln ∣ tan (3 x) - 3 ∣ + C

What is the integral of (10sec^2(3x))/(tan(3x)-3) ?
The integral of (10sec^2(3x))/(tan(3x)-3) is 10/3 ln|tan(3x)-3|+C