What is the integral of (sec^2(5x))/(1-tan(5x)) ?

The integral of (sec^2(5x))/(1-tan(5x)) is -1/5 ln|1-tan(5x)|+C

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integral of (sec^2(5x))/(1-tan(5x))

\int \frac{sec ^{2} ( 5 x )}{1 - tan ( 5 x )} d x

- \frac{1}{5} ln ∣ 1 - tan (5 x) ∣ + C

Solution steps

\int \frac{sec ^{2} ( 5 x )}{1 - tan ( 5 x )} d x

Apply u-substitution

= \int - \frac{1}{5 u} d u

Take the constant out:

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

= - \frac{1}{5} \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= - \frac{1}{5} ln ∣ u ∣

Substitute back

u = 1 - tan (5 x)

= - \frac{1}{5} ln ∣ 1 - tan (5 x) ∣

Add a constant to the solution

= - \frac{1}{5} ln ∣ 1 - tan (5 x) ∣ + C

What is the integral of (sec^2(5x))/(1-tan(5x)) ?
The integral of (sec^2(5x))/(1-tan(5x)) is -1/5 ln|1-tan(5x)|+C