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

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

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

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

2 tan (x) + 1 + C

Solution steps

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

= 2 tan (x) + 1

Add a constant to the solution

= 2 tan (x) + 1 + C

\int \frac{1}{x} 1 + \frac{1}{x} d x

\int (3 x^{2} + 7 x + \frac{1}{x ^{3}}) d x

\int 2 x^{14} d x

\int \frac{1}{x ^{3} + x ^{2} + 2 x + 2} d x

\int \frac{u - 1}{u} d u

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