What is the integral of (x^2)/(sqrt((x^2-1)^3)) ?

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

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integral of (x^2)/(sqrt((x^2-1)^3))

\int \frac{x ^{2}}{( x ^{2} - 1 ) ^{3}} d x

- \frac{x}{x ^{2} - 1} + ln x^{2} - 1 + x + C

Solution steps

\int \frac{x ^{2}}{( x ^{2} - 1 ) ^{3}} d x

(x^{2} - 1)^{3} = (x^{2} - 1)^{\frac{3}{2}},

assuming

(x^{2} - 1) \geq 0

= \int \frac{x ^{2}}{( x ^{2} - 1 ) ^{\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 (x)

= - csc (arcsec (x)) - (- ln ∣ tan (arcsec (x)) + sec (arcsec (x)) ∣)

Simplify

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

= - \frac{x}{x ^{2} - 1} + ln x^{2} - 1 + x

Add a constant to the solution

= - \frac{x}{x ^{2} - 1} + ln x^{2} - 1 + x + C

n = 1 \sum \infty \frac{n}{n + 9}

f (x) = arcsec (5 x)

\int x os (2 x) d x

\frac{d}{d x} (a sin (b x + c) + d)

\int 5 e^{a x} d x

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