What is the integral of (sqrt(81x^2-49))/(x^3) ?

The integral of (sqrt(81x^2-49))/(x^3) is -(sqrt(81x^2-49))/(2x^2)+81/14 arcsec(9/7 x)+C

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integral of (sqrt(81x^2-49))/(x^3)

\int \frac{81 x ^{2} - 49}{x ^{3}} d x

- \frac{81 x ^{2} - 49}{2 x ^{2}} + \frac{81}{14} arcsec (\frac{9}{7} x) + C

Solution steps

\int \frac{81 x ^{2} - 49}{x ^{3}} d x

Apply Integration By Parts

: - \frac{81 x ^{2} - 49}{2 x ^{2}} - \int - \frac{81}{2 x 81 x ^{2} - 49} d x

= - \frac{81 x ^{2} - 49}{2 x ^{2}} - \int - \frac{81}{2 x 81 x ^{2} - 49} d x

\int - \frac{81}{2 x 81 x ^{2} - 49} d x = - \frac{81}{14} arcsec (\frac{9}{7} x)

= - \frac{81 x ^{2} - 49}{2 x ^{2}} - (- \frac{81}{14} arcsec (\frac{9}{7} x))

Simplify

= - \frac{81 x ^{2} - 49}{2 x ^{2}} + \frac{81}{14} arcsec (\frac{9}{7} x)

Add a constant to the solution

= - \frac{81 x ^{2} - 49}{2 x ^{2}} + \frac{81}{14} arcsec (\frac{9}{7} x) + C

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What is the integral of (sqrt(81x^2-49))/(x^3) ?
The integral of (sqrt(81x^2-49))/(x^3) is -(sqrt(81x^2-49))/(2x^2)+81/14 arcsec(9/7 x)+C