What is the integral of [(sqrt(x^2-9))/x ] ?

The integral of [(sqrt(x^2-9))/x ] is -3arcsec(1/3 x)+sqrt(x^2-9)+C

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integral of [(sqrt(x^2-9))/x ]

\int [\frac{x ^{2} - 9}{x}] d x

- 3 arcsec (\frac{1}{3} x) + x^{2} - 9 + C

Solution steps

\int \frac{x ^{2} - 9}{x} d x

Apply Trigonometric Substitution

: \int 3 tan^{2} (u) d u

= \int 3 tan^{2} (u) d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 3 \cdot \int tan^{2} (u) d u

Rewrite using trig identities

= 3 \cdot \int - 1 + sec^{2} (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

= 3 (- \int 1 d u + \int sec^{2} (u) d u)

\int 1 d u = u

\int sec^{2} (u) d u = tan (u)

= 3 (- u + tan (u))

Substitute back

u = arcsec (\frac{1}{3} x)

= 3 (- arcsec (\frac{1}{3} x) + tan (arcsec (\frac{1}{3} x)))

Simplify

3 (- arcsec (\frac{1}{3} x) + tan (arcsec (\frac{1}{3} x))) : - 3 arcsec (\frac{1}{3} x) + x^{2} - 9

= - 3 arcsec (\frac{1}{3} x) + x^{2} - 9

Add a constant to the solution

= - 3 arcsec (\frac{1}{3} x) + x^{2} - 9 + C

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What is the integral of [(sqrt(x^2-9))/x ] ?
The integral of [(sqrt(x^2-9))/x ] is -3arcsec(1/3 x)+sqrt(x^2-9)+C