What is the integral of (1-x^{-2})sqrt(x+x^{-1)} ?

The integral of (1-x^{-2})sqrt(x+x^{-1)} is 2/3 ((x^2+1)/x)^{3/2}+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

integral of (1-x^{-2})sqrt(x+x^{-1)}

\int (1 - x^{- 2}) x + x^{- 1} d x

\frac{2}{3} (\frac{x ^{2} + 1}{x})^{\frac{3}{2}} + C

Solution steps

\int (1 - x^{- 2}) x + x^{- 1} d x

Apply u-substitution

= \int u d u

Apply the Power Rule

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

Substitute back

u = x + x^{- 1}

= \frac{2}{3} (x + x^{- 1})^{\frac{3}{2}}

(x + x^{- 1})^{\frac{3}{2}} = (\frac{x ^{2} + 1}{x})^{\frac{3}{2}}

= \frac{2}{3} (\frac{x ^{2} + 1}{x})^{\frac{3}{2}}

Add a constant to the solution

= \frac{2}{3} (\frac{x ^{2} + 1}{x})^{\frac{3}{2}} + C

\int \frac{1}{x + 1 ^{5}} d x

\int (3 z^{2} + 1) arctan (z)

\int 2 x (x^{3} + 1) d x

\int 3 t e^{- s t} d t

\int - sin (x) + e^{x} d x

What is the integral of (1-x^{-2})sqrt(x+x^{-1)} ?
The integral of (1-x^{-2})sqrt(x+x^{-1)} is 2/3 ((x^2+1)/x)^{3/2}+C