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

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

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

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

ln ∣ tan (x) + sec (x) ∣ + C

Solution steps

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

Simplify

\frac{1}{cos ^{2} ( x ) 1 + tan ^{2} ( x )} : sec^{2} (x)

= \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

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