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

The integral of (sqrt(tan(x)+1))/(cos^2(x)) is 2/3 (tan(x)+1)^{3/2}+C

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

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

\frac{2}{3} (tan (x) + 1)^{\frac{3}{2}} + C

Solution steps

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

Rewrite using trig identities

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

Apply u-substitution

= \int u d u

Apply the Power Rule

= \frac{2}{3} u^{\frac{3}{2}}

Substitute back

u = tan (x) + 1

= \frac{2}{3} (tan (x) + 1)^{\frac{3}{2}}

Add a constant to the solution

= \frac{2}{3} (tan (x) + 1)^{\frac{3}{2}} + C

What is the integral of (sqrt(tan(x)+1))/(cos^2(x)) ?
The integral of (sqrt(tan(x)+1))/(cos^2(x)) is 2/3 (tan(x)+1)^{3/2}+C