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

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

\int \frac{1}{x ( 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

= \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

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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