What is the integral of (sqrt((x-2)^2-16))/(x-2) ?

The integral of (sqrt((x-2)^2-16))/(x-2) is -4arcsec(1/4 (x-2))+sqrt(x^2-4x-12)+C

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integral of (sqrt((x-2)^2-16))/(x-2)

\int \frac{( x - 2 ) ^{2} - 16}{x - 2} d x

- 4 arcsec (\frac{1}{4} (x - 2)) + x^{2} - 4 x - 12 + C

Solution steps

\int \frac{( x - 2 ) ^{2} - 16}{x - 2} d x

Apply u-substitution

: \int \frac{u ^{2} - 16}{u} d u

= \int \frac{u ^{2} - 16}{u} d u

Apply Trigonometric Substitution

: \int 4 tan^{2} (v) d v

= \int 4 tan^{2} (v) d v

Take the constant out:

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

= 4 \cdot \int tan^{2} (v) d v

Rewrite using trig identities

= 4 \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

= 4 (- \int 1 d v + \int sec^{2} (v) d v)

\int 1 d v = v

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

= 4 (- v + tan (v))

Substitute back

= 4 (- arcsec (\frac{1}{4} (x - 2)) + tan (arcsec (\frac{1}{4} (x - 2))))

Simplify

4 (- arcsec (\frac{1}{4} (x - 2)) + tan (arcsec (\frac{1}{4} (x - 2)))) : - 4 arcsec (\frac{1}{4} (x - 2)) + x^{2} - 4 x - 12

= - 4 arcsec (\frac{1}{4} (x - 2)) + x^{2} - 4 x - 12

Add a constant to the solution

= - 4 arcsec (\frac{1}{4} (x - 2)) + x^{2} - 4 x - 12 + C

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What is the integral of (sqrt((x-2)^2-16))/(x-2) ?
The integral of (sqrt((x-2)^2-16))/(x-2) is -4arcsec(1/4 (x-2))+sqrt(x^2-4x-12)+C