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

The integral of 1/(x^3sqrt(x^2-a^2)) is 1/(2a^3)(arcsec(x/a)+1/2 sin(2arcsec(x/a)))+C

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

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

\frac{1}{2 a ^{3}} (arcsec (\frac{x}{a}) + \frac{1}{2} sin (2 arcsec (\frac{x}{a}))) + C

Solution steps

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

Apply Trigonometric Substitution

= \int \frac{1}{a ^{3} sec ^{2} ( u )} d u

Take the constant out:

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

= \frac{1}{a ^{3}} \cdot \int \frac{1}{sec ^{2} ( u )} d u

Rewrite using trig identities

= \frac{1}{a ^{3}} \cdot \int \frac{1 + cos ( 2 u )}{2} d u

Take the constant out:

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

= \frac{1}{a ^{3}} \cdot \frac{1}{2} \cdot \int 1 + cos (2 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

= \frac{1}{a ^{3}} \cdot \frac{1}{2} (\int 1 d u + \int cos (2 u) d u)

\int 1 d u = u

\int cos (2 u) d u = \frac{1}{2} sin (2 u)

= \frac{1}{a ^{3}} \cdot \frac{1}{2} (u + \frac{1}{2} sin (2 u))

Substitute back

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

= \frac{1}{a ^{3}} \cdot \frac{1}{2} (arcsec (\frac{1}{a} x) + \frac{1}{2} sin (2 arcsec (\frac{1}{a} x)))

Simplify

\frac{1}{a ^{3}} \cdot \frac{1}{2} (arcsec (\frac{1}{a} x) + \frac{1}{2} sin (2 arcsec (\frac{1}{a} x))) : \frac{1}{2 a ^{3}} (arcsec (\frac{x}{a}) + \frac{1}{2} sin (2 arcsec (\frac{x}{a})))

= \frac{1}{2 a ^{3}} (arcsec (\frac{x}{a}) + \frac{1}{2} sin (2 arcsec (\frac{x}{a})))

Add a constant to the solution

= \frac{1}{2 a ^{3}} (arcsec (\frac{x}{a}) + \frac{1}{2} sin (2 arcsec (\frac{x}{a}))) + C

What is the integral of 1/(x^3sqrt(x^2-a^2)) ?
The integral of 1/(x^3sqrt(x^2-a^2)) is 1/(2a^3)(arcsec(x/a)+1/2 sin(2arcsec(x/a)))+C