What is the integral of+(sqrt(x^2-81))/(x^3) ?

The integral of+(sqrt(x^2-81))/(x^3) is -(sqrt(x^2-81))/(2x^2)+1/18 arcsec(1/9 x)+C

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integral of+(sqrt(x^2-81))/(x^3)

\int + \frac{x ^{2} - 81}{x ^{3}} d x

- \frac{x ^{2} - 81}{2 x ^{2}} + \frac{1}{18} arcsec (\frac{1}{9} x) + C

Solution steps

\int \frac{x ^{2} - 81}{x ^{3}} d x

Apply Integration By Parts

: - \frac{x ^{2} - 81}{2 x ^{2}} - \int - \frac{1}{2 x x ^{2} - 81} d x

= - \frac{x ^{2} - 81}{2 x ^{2}} - \int - \frac{1}{2 x x ^{2} - 81} d x

\int - \frac{1}{2 x x ^{2} - 81} d x = - \frac{1}{18} arcsec (\frac{1}{9} x)

= - \frac{x ^{2} - 81}{2 x ^{2}} - (- \frac{1}{18} arcsec (\frac{1}{9} x))

Simplify

= - \frac{x ^{2} - 81}{2 x ^{2}} + \frac{1}{18} arcsec (\frac{1}{9} x)

Add a constant to the solution

= - \frac{x ^{2} - 81}{2 x ^{2}} + \frac{1}{18} arcsec (\frac{1}{9} x) + C

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What is the integral of+(sqrt(x^2-81))/(x^3) ?
The integral of+(sqrt(x^2-81))/(x^3) is -(sqrt(x^2-81))/(2x^2)+1/18 arcsec(1/9 x)+C