What is the integral of sec^2(x)ln(cos(x)) ?

The integral of sec^2(x)ln(cos(x)) is ln(cos(x))tan(x)-x+tan(x)+C

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integral of sec^2(x)ln(cos(x))

\int sec^{2} (x) ln (cos (x)) d x

ln (cos (x)) tan (x) - x + tan (x) + C

Solution steps

\int sec^{2} (x) ln (cos (x)) d x

Rewrite using trig identities

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

Apply Integration By Parts

: ln (cos (x)) tan (x) - \int - tan^{2} (x) d x

= ln (cos (x)) tan (x) - \int - tan^{2} (x) d x

\int - tan^{2} (x) d x = x - tan (x)

= ln (cos (x)) tan (x) - (x - tan (x))

Simplify

= ln (cos (x)) tan (x) - x + tan (x)

Add a constant to the solution

= ln (cos (x)) tan (x) - x + tan (x) + C

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What is the integral of sec^2(x)ln(cos(x)) ?
The integral of sec^2(x)ln(cos(x)) is ln(cos(x))tan(x)-x+tan(x)+C