What is the integral of sqrt(1+tan^2(x)) ?

The integral of sqrt(1+tan^2(x)) is ln|tan(x)+sec(x)|+C

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integral of sqrt(1+tan^2(x))

integral

1 + tan^{2} (x)

ln ∣ tan (x) + sec (x) ∣ + C

Solution steps

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

Rewrite using trig identities

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

sec^{2} (x) = (sec (x)),

assuming

sec (x) \geq 0

= \int sec (x) d x

Use the common integral:

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

= ln ∣ tan (x) + sec (x) ∣

Add a constant to the solution

= ln ∣ tan (x) + sec (x) ∣ + C

What is the integral of sqrt(1+tan^2(x)) ?
The integral of sqrt(1+tan^2(x)) is ln|tan(x)+sec(x)|+C