What is the integral of (sqrt(5+2tan(x)))/(cos^2(x)) ?

The integral of (sqrt(5+2tan(x)))/(cos^2(x)) is 1/3 (5+2tan(x))^{3/2}+C

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integral of (sqrt(5+2tan(x)))/(cos^2(x))

\int \frac{5 + 2 tan ( x )}{cos ^{2} ( x )} d x

\frac{1}{3} (5 + 2 tan (x))^{\frac{3}{2}} + C

Solution steps

\int \frac{5 + 2 tan ( x )}{cos ^{2} ( x )} d x

Rewrite using trig identities

= \int 5 + 2 tan (x) sec^{2} (x) d x

Apply u-substitution

: \int u^{2} d u

= \int u^{2} d u

Apply the Power Rule

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

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

Substitute back

u = 5 + 2 tan (x)

= \frac{( 5 + 2 tan ( x ) ) ^{3}}{3}

Simplify

\frac{( 5 + 2 tan ( x ) ) ^{3}}{3} : \frac{1}{3} (5 + 2 tan (x))^{\frac{3}{2}}

= \frac{1}{3} (5 + 2 tan (x))^{\frac{3}{2}}

Add a constant to the solution

= \frac{1}{3} (5 + 2 tan (x))^{\frac{3}{2}} + C

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What is the integral of (sqrt(5+2tan(x)))/(cos^2(x)) ?
The integral of (sqrt(5+2tan(x)))/(cos^2(x)) is 1/3 (5+2tan(x))^{3/2}+C