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

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

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

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

2 1 + tan (x) + C

Solution steps

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

Rewrite using trig identities

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

Apply u-substitution

: \int 2 d u

= \int 2 d u

Integral of a constant:

\int a d x = a x

= 2 u

Substitute back

u = 1 + tan (x)

= 2 1 + tan (x)

Add a constant to the solution

= 2 1 + tan (x) + C

\int \frac{arctan ^{3} ( x )}{3 x ^{2} + 3} d x

\int arctan (tan (t)) d t

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

\int \frac{2 x ^{2}}{4 - x ^{2}} d x

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