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

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

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

\int cos (x) ln (tan (x)) d x

ln (tan (x)) sin (x) - ln ∣ tan (x) + sec (x) ∣ + C

Solution steps

\int cos (x) ln (tan (x)) d x

Apply Integration By Parts

: ln (tan (x)) sin (x) - \int sec (x) d x

= ln (tan (x)) sin (x) - \int sec (x) d x

\int sec (x) d x = ln ∣ tan (x) + sec (x) ∣

= ln (tan (x)) sin (x) - ln ∣ tan (x) + sec (x) ∣

Add a constant to the solution

= ln (tan (x)) sin (x) - ln ∣ tan (x) + sec (x) ∣ + C

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