What is the integral of (sec^4(x))/(sqrt(tan(x))) ?

The integral of (sec^4(x))/(sqrt(tan(x))) is 2sqrt(tan(x))+2/5 tan^{5/2}(x)+C

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integral of (sec^4(x))/(sqrt(tan(x)))

\int \frac{sec ^{4} ( x )}{tan ( x )} d x

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

Solution steps

\int \frac{sec ^{4} ( x )}{tan ( x )} d x

Apply u-substitution

: \int \frac{1 + u ^{2}}{u} d u

= \int \frac{1 + u ^{2}}{u} d u

Expand

\frac{1 + u ^{2}}{u} : \frac{1}{u} + u^{\frac{3}{2}}

= \int \frac{1}{u} + u^{\frac{3}{2}} 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 \frac{1}{u} d u + \int u^{\frac{3}{2}} d u

\int \frac{1}{u} d u = 2 u

\int u^{\frac{3}{2}} d u = \frac{2}{5} u^{\frac{5}{2}}

= 2 u + \frac{2}{5} u^{\frac{5}{2}}

Substitute back

u = tan (x)

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

Add a constant to the solution

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

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What is the integral of (sec^4(x))/(sqrt(tan(x))) ?
The integral of (sec^4(x))/(sqrt(tan(x))) is 2sqrt(tan(x))+2/5 tan^{5/2}(x)+C