What is the integral of 1/(x*(x^2-1)^{1/2)} ?

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

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

integral of 1/(x*(x^2-1)^{1/2)}

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

arcsec (x) + C

Solution steps

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

Apply Integration By Parts

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

\int - \frac{2 x ^{2} - 1}{x ( x ^{2} - 1 ) ^{\frac{3}{2}}} d x = \frac{1}{( x ^{2} - 1 ) ^{\frac{1}{2}}} - arcsec (x)

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

Simplify

= arcsec (x)

Add a constant to the solution

= arcsec (x) + C

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