What is the integral of (sqrt(5-3tan(x)))/(cos^2(x)) ?

The integral of (sqrt(5-3tan(x)))/(cos^2(x)) is -2/9 (5-3tan(x))^{3/2}+C

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integral of (sqrt(5-3tan(x)))/(cos^2(x))

\int \frac{5 - 3 tan ( x )}{cos ^{2} ( x )} d x

- \frac{2}{9} (5 - 3 tan (x))^{\frac{3}{2}} + C

Solution steps

\int \frac{5 - 3 tan ( x )}{cos ^{2} ( x )} d x

Use the basic trigonometric identity:

\frac{1}{cos ( x )} = sec (x)

= \int sec^{2} (x) 5 - 3 tan (x) d x

Apply u-substitution

: \int - \frac{u}{3} d u

= \int - \frac{u}{3} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= - \frac{1}{3} \cdot \int u d u

Apply the Power Rule

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

= - \frac{1}{3} \cdot \frac{2}{3} u^{\frac{3}{2}}

Substitute back

u = 5 - 3 tan (x)

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

Simplify

- \frac{1}{3} \cdot \frac{2}{3} (5 - 3 tan (x))^{\frac{3}{2}} : - \frac{2}{9} (5 - 3 tan (x))^{\frac{3}{2}}

= - \frac{2}{9} (5 - 3 tan (x))^{\frac{3}{2}}

Add a constant to the solution

= - \frac{2}{9} (5 - 3 tan (x))^{\frac{3}{2}} + C

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What is the integral of (sqrt(5-3tan(x)))/(cos^2(x)) ?
The integral of (sqrt(5-3tan(x)))/(cos^2(x)) is -2/9 (5-3tan(x))^{3/2}+C