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

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

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

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

arctan (x^{2} - 1) - arcsec (x) - \frac{1}{2} sin (2 arcsec (x)) + C

Solution steps

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

Expand

\frac{x ^{2} - 2}{x ^{3} x ^{2} - 1} : \frac{1}{x x ^{2} - 1} - \frac{2}{x ^{3} x ^{2} - 1}

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

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

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

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

\int \frac{2}{x ^{3} x ^{2} - 1} d x = arcsec (x) + \frac{1}{2} sin (2 arcsec (x))

= arctan (x^{2} - 1) - (arcsec (x) + \frac{1}{2} sin (2 arcsec (x)))

Simplify

= arctan (x^{2} - 1) - arcsec (x) - \frac{1}{2} sin (2 arcsec (x))

Add a constant to the solution

= arctan (x^{2} - 1) - arcsec (x) - \frac{1}{2} sin (2 arcsec (x)) + C

\int \frac{x ^{3} - 4 x + 3}{x + 5} d x

\int \frac{c}{r ^{2}} d r

\int r r + 3 d r

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

\int \frac{3 sin ( 2 x )}{cos ( 2 x ) + 2} d x

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