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

The integral of 1/(cos^2(θ)sqrt(1+tan(θ))) is 2sqrt(1+tan(θ))+C

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

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

2 1 + tan (θ) + C

Solution steps

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

Rewrite using trig identities

= \int \frac{sec ^{2} ( θ )}{1 + tan ( θ )} d θ

Apply u-substitution

= \int 2 d u

Integral of a constant:

\int a d x = a x

= 2 u

Substitute back

u = 1 + tan (θ)

= 2 1 + tan (θ)

Add a constant to the solution

= 2 1 + tan (θ) + C

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