What is the integral of (sqrt(x^2-2x-8))/(x-1) ?

The integral of (sqrt(x^2-2x-8))/(x-1) is -3arcsec(1/3 (x-1))+sqrt(x^2-2x-8)+C

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integral of (sqrt(x^2-2x-8))/(x-1)

\int \frac{x ^{2} - 2 x - 8}{x - 1} d x

- 3 arcsec (\frac{1}{3} (x - 1)) + x^{2} - 2 x - 8 + C

Solution steps

\int \frac{x ^{2} - 2 x - 8}{x - 1} d x

Apply u-substitution

: \int \frac{u ^{2} - 9}{u} d u

= \int \frac{u ^{2} - 9}{u} d u

Apply Trigonometric Substitution

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

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

Take the constant out:

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

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

Rewrite using trig identities

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

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 v + \int sec^{2} (v) d v)

\int 1 d v = v

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

= 3 (- v + tan (v))

Substitute back

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

Simplify

3 (- arcsec (\frac{1}{3} (x - 1)) + tan (arcsec (\frac{1}{3} (x - 1)))) : - 3 arcsec (\frac{1}{3} (x - 1)) + x^{2} - 2 x - 8

= - 3 arcsec (\frac{1}{3} (x - 1)) + x^{2} - 2 x - 8

Add a constant to the solution

= - 3 arcsec (\frac{1}{3} (x - 1)) + x^{2} - 2 x - 8 + C

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What is the integral of (sqrt(x^2-2x-8))/(x-1) ?
The integral of (sqrt(x^2-2x-8))/(x-1) is -3arcsec(1/3 (x-1))+sqrt(x^2-2x-8)+C