What is the integral of (x^2)/((x^2-a^2)^{3/2)} ?

The integral of (x^2)/((x^2-a^2)^{3/2)} is -x/((x^2-a^2)^{1/2)}+ln((|(x^2-a^2)^{1/2}+x|)/(|a|))+C

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integral of (x^2)/((x^2-a^2)^{3/2)}

\int \frac{x ^{2}}{( x ^{2} - a ^{2} ) ^{\frac{3}{2}}} d x

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

Solution steps

\int \frac{x ^{2}}{( x ^{2} - a ^{2} ) ^{\frac{3}{2}}} d x

Apply Trigonometric Substitution

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

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

Rewrite using trig identities

= \int \frac{csc ^{2} ( u )}{cos ( u )} d u

Apply Integration By Parts

: - csc (u) - \int - sec (u) d u

= - csc (u) - \int - sec (u) d u

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

= - csc (u) - (- ln ∣ tan (u) + sec (u) ∣)

Substitute back

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

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

Simplify

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

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

Add a constant to the solution

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

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What is the integral of (x^2)/((x^2-a^2)^{3/2)} ?
The integral of (x^2)/((x^2-a^2)^{3/2)} is -x/((x^2-a^2)^{1/2)}+ln((|(x^2-a^2)^{1/2}+x|)/(|a|))+C