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

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

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

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

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

Solution steps

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

Rewrite using trig identities

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

Apply u-substitution

: \int u d u

= \int u d u

Apply the Power Rule

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

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

Substitute back

u = 1 + tan (x)

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

Add a constant to the solution

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

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