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

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

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

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

- ln ∣ ln (cos (x)) ∣ + C

Solution steps

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

Apply u-substitution

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

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

Take the constant out:

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

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

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= - ln ∣ u ∣

Substitute back

u = ln (cos (x))

= - ln ∣ ln (cos (x)) ∣

Add a constant to the solution

= - ln ∣ ln (cos (x)) ∣ + C

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\int e^{x} (1 - \frac{e ^{- x}}{x}) d x

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