What is the integral of tan^3(x)sqrt(sec(x)) ?

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

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integral of tan^3(x)sqrt(sec(x))

\int tan^{3} (x) sec (x) d x

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

Solution steps

\int tan^{3} (x) sec (x) d x

Rewrite using trig identities

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

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

Add a constant to the solution

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

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