What is the integral of (sqrt(25x^2-36))/(x^3) ?

The integral of (sqrt(25x^2-36))/(x^3) is -(sqrt(25x^2-36))/(2x^2)+25/12 arcsec(5/6 x)+C

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integral of (sqrt(25x^2-36))/(x^3)

\int \frac{25 x ^{2} - 36}{x ^{3}} d x

- \frac{25 x ^{2} - 36}{2 x ^{2}} + \frac{25}{12} arcsec (\frac{5}{6} x) + C

Solution steps

\int \frac{25 x ^{2} - 36}{x ^{3}} d x

Apply Integration By Parts

: - \frac{25 x ^{2} - 36}{2 x ^{2}} - \int - \frac{25}{2 x 25 x ^{2} - 36} d x

= - \frac{25 x ^{2} - 36}{2 x ^{2}} - \int - \frac{25}{2 x 25 x ^{2} - 36} d x

\int - \frac{25}{2 x 25 x ^{2} - 36} d x = - \frac{25}{12} arcsec (\frac{5}{6} x)

= - \frac{25 x ^{2} - 36}{2 x ^{2}} - (- \frac{25}{12} arcsec (\frac{5}{6} x))

Simplify

= - \frac{25 x ^{2} - 36}{2 x ^{2}} + \frac{25}{12} arcsec (\frac{5}{6} x)

Add a constant to the solution

= - \frac{25 x ^{2} - 36}{2 x ^{2}} + \frac{25}{12} arcsec (\frac{5}{6} x) + C

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What is the integral of (sqrt(25x^2-36))/(x^3) ?
The integral of (sqrt(25x^2-36))/(x^3) is -(sqrt(25x^2-36))/(2x^2)+25/12 arcsec(5/6 x)+C