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

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

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

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

\frac{2}{3} sec^{\frac{3}{2}} (x) + \frac{2}{sec ( x )} + C

Solution steps

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

Apply u-substitution

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

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

Expand

\frac{u ^{2} - 1}{u u} : u - \frac{1}{u u}

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

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

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

= \frac{2}{3} u^{\frac{3}{2}} - (- \frac{2}{u})

Substitute back

u = sec (x)

= \frac{2}{3} sec^{\frac{3}{2}} (x) - (- \frac{2}{sec ( x )})

Simplify

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

Add a constant to the solution

= \frac{2}{3} sec^{\frac{3}{2}} (x) + \frac{2}{sec ( x )} + C

\int - u^{2} d u

\int_{0}^{7} \frac{1}{u ^{2} - 49} d u

\frac{d}{d x} (\frac{2000}{x ^{2}})

\frac{d x}{d t} = x^{2} t^{2}

\frac{d}{d x} (\frac{x ^{3} + 1}{x})

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