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

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

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

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

- \frac{5}{2} ln ∣ 1 - tan (x) ∣ + C

Solution steps

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

Take the constant out:

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

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

Apply u-substitution

: \int - \frac{1}{u} d u

= \frac{5}{2} \cdot \int - \frac{1}{u} d u

Take the constant out:

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

= \frac{5}{2} (- \int \frac{1}{u} d u)

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= \frac{5}{2} (- ln ∣ u ∣)

Substitute back

u = 1 - tan (x)

= \frac{5}{2} (- ln ∣ 1 - tan (x) ∣)

Simplify

= - \frac{5}{2} ln ∣ 1 - tan (x) ∣

Add a constant to the solution

= - \frac{5}{2} ln ∣ 1 - tan (x) ∣ + C

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