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

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

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

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

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

Solution steps

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

Use the basic trigonometric identity:

\frac{1}{cos ( x )} = sec (x)

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