What is the integral of (2-sqrt(tan(x)))^2sec^2(x) ?

The integral of (2-sqrt(tan(x)))^2sec^2(x) is 4tan(x)-8/3 tan^{3/2}(x)+(tan^2(x))/2+C

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integral of (2-sqrt(tan(x)))^2sec^2(x)

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

4 tan (x) - \frac{8}{3} tan^{\frac{3}{2}} (x) + \frac{tan ^{2} ( x )}{2} + C

Solution steps

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

Apply u-substitution

: \int (2 - u)^{2} d u

= \int (2 - u)^{2} d u

Expand

(2 - u)^{2} : 4 - 4 u + u

= \int 4 - 4 u + u d u

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int 4 d u - \int 4 u d u + \int u d u

\int 4 d u = 4 u

\int 4 u d u = \frac{8}{3} u^{\frac{3}{2}}

\int u d u = \frac{u ^{2}}{2}

= 4 u - \frac{8}{3} u^{\frac{3}{2}} + \frac{u ^{2}}{2}

Substitute back

u = tan (x)

= 4 tan (x) - \frac{8}{3} tan^{\frac{3}{2}} (x) + \frac{tan ^{2} ( x )}{2}

Add a constant to the solution

= 4 tan (x) - \frac{8}{3} tan^{\frac{3}{2}} (x) + \frac{tan ^{2} ( x )}{2} + C

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What is the integral of (2-sqrt(tan(x)))^2sec^2(x) ?
The integral of (2-sqrt(tan(x)))^2sec^2(x) is 4tan(x)-8/3 tan^{3/2}(x)+(tan^2(x))/2+C