What is the integral of (ln(1+tan(x)))/(cos^2(x)) ?

The integral of (ln(1+tan(x)))/(cos^2(x)) is ln(1+tan(x))+ln(1+tan(x))tan(x)-1-tan(x)+C

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integral of (ln(1+tan(x)))/(cos^2(x))

\int \frac{ln ( 1 + tan ( x ) )}{cos ^{2} ( x )} d x

ln (1 + tan (x)) + ln (1 + tan (x)) tan (x) - 1 - tan (x) + C

Solution steps

\int \frac{ln ( 1 + tan ( x ) )}{cos ^{2} ( x )} d x

Use the basic trigonometric identity:

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

= \int sec^{2} (x) ln (1 + tan (x)) d x

Apply u-substitution

: \int ln (1 + u) d u

= \int ln (1 + u) d u

Apply u-substitution

: \int ln (v) d v

= \int ln (v) d v

Apply Integration By Parts

: v ln (v) - \int 1 d v

= v ln (v) - \int 1 d v

\int 1 d v = v

= v ln (v) - v

Substitute back

= (1 + tan (x)) ln (1 + tan (x)) - (1 + tan (x))

Simplify

(1 + tan (x)) ln (1 + tan (x)) - (1 + tan (x)) : ln (1 + tan (x)) + ln (1 + tan (x)) tan (x) - 1 - tan (x)

= ln (1 + tan (x)) + ln (1 + tan (x)) tan (x) - 1 - tan (x)

Add a constant to the solution

= ln (1 + tan (x)) + ln (1 + tan (x)) tan (x) - 1 - tan (x) + C

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What is the integral of (ln(1+tan(x)))/(cos^2(x)) ?
The integral of (ln(1+tan(x)))/(cos^2(x)) is ln(1+tan(x))+ln(1+tan(x))tan(x)-1-tan(x)+C