What is the integral of (x^2-a^2)^{1/2} ?

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

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integral of (x^2-a^2)^{1/2}

integral

(x^{2} - a^{2})^{\frac{1}{2}}

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

Solution steps

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

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

Add a constant to the solution

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

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