What is the integral of (tan(2x))/(ln(cos(2x))) ?

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

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integral of (tan(2x))/(ln(cos(2x)))

\int \frac{tan ( 2 x )}{ln ( cos ( 2 x ) )} d x

- \frac{1}{2} ln ∣ ln (cos (2 x)) ∣ + C

Solution steps

\int \frac{tan ( 2 x )}{ln ( cos ( 2 x ) )} d x

Apply u-substitution

: \int - \frac{1}{2 u} d u

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

Take the constant out:

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

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

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= - \frac{1}{2} ln ∣ u ∣

Substitute back

u = ln (cos (2 x))

= - \frac{1}{2} ln ∣ ln (cos (2 x)) ∣

Add a constant to the solution

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

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