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

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

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

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

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

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

Solution steps

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

Rewrite using trig identities

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

Apply Integration By Parts

= 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

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