What is the integral of 1/((x^2sqrt(x^2-1))) ?

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

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integral of 1/((x^2sqrt(x^2-1)))

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

\frac{x ^{2} - 1}{x} + C

Solution steps

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

Apply Trigonometric Substitution

= \int cos (u) d u

Use the common integral:

\int cos (u) d u = sin (u)

= sin (u)

Substitute back

u = arcsec (x)

= sin (arcsec (x))

Use the following identity:

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

= \frac{x ^{2} - 1}{x}

Add a constant to the solution

= \frac{x ^{2} - 1}{x} + C

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