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

The integral of sqrt(x^2-a^2) is 1/2 xsqrt(x^2-a^2)-(a^2)/2 ln((|x+sqrt(x^2-a^2)|)/(|a|))+C

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integral of sqrt(x^2-a^2)

integral

x^{2} - a^{2}

\frac{1}{2} x x^{2} - a^{2} - \frac{a ^{2}}{2} ln (\frac{x + x ^{2} - a ^{2}}{∣ a ∣}) + C

Solution steps

\int x^{2} - a^{2} d x

Apply Trigonometric Substitution

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

Take the constant out:

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

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

Rewrite using trig identities

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

Expand

(- 1 + sec^{2} (u)) sec (u) : - sec (u) + sec^{3} (u)

= a^{2} \cdot \int - sec (u) + sec^{3} (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

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

\int sec (u) d u = ln ∣ tan (u) + sec (u) ∣

\int sec^{3} (u) d u = \frac{1}{2} sec (u) tan (u) + \frac{1}{2} ln ∣ tan (u) + sec (u) ∣

= a^{2} (- ln ∣ tan (u) + sec (u) ∣ + \frac{1}{2} sec (u) tan (u) + \frac{1}{2} ln ∣ tan (u) + sec (u) ∣)

Substitute back

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

= a^{2} (- ln tan (arcsec (\frac{1}{a} x)) + sec (arcsec (\frac{1}{a} x)) + \frac{1}{2} sec (arcsec (\frac{1}{a} x)) tan (arcsec (\frac{1}{a} x)) + \frac{1}{2} ln tan (arcsec (\frac{1}{a} x)) + sec (arcsec (\frac{1}{a} x)))

Simplify

a^{2} (- ln tan (arcsec (\frac{1}{a} x)) + sec (arcsec (\frac{1}{a} x)) + \frac{1}{2} sec (arcsec (\frac{1}{a} x)) tan (arcsec (\frac{1}{a} x)) + \frac{1}{2} ln tan (arcsec (\frac{1}{a} x)) + sec (arcsec (\frac{1}{a} x))) : \frac{1}{2} x x^{2} - a^{2} - \frac{a ^{2}}{2} ln (\frac{x + x ^{2} - a ^{2}}{∣ a ∣})

= \frac{1}{2} x x^{2} - a^{2} - \frac{a ^{2}}{2} ln (\frac{x + x ^{2} - a ^{2}}{∣ a ∣})

Add a constant to the solution

= \frac{1}{2} x x^{2} - a^{2} - \frac{a ^{2}}{2} ln (\frac{x + x ^{2} - a ^{2}}{∣ a ∣}) + C

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