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

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

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 7 x e^{- 5 x^{2}} d x

\int (cos (x) + 2^{x}) d x

\int 2 (cos (x))^{2} d x

\int t e^{3 t^{2} - 1} d t

\int \frac{x ^{2} + 2 x - 4}{( x + 3 ) ( x + 4 ) ( x - 5 )} d 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