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

The integral of (sqrt(x^2-25))/(x^3) is -(sqrt(x^2-25))/(2x^2)+1/10 arcsec(1/5 x)+C

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

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

- \frac{x ^{2} - 25}{2 x ^{2}} + \frac{1}{10} arcsec (\frac{1}{5} x) + C

Solution steps

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

Apply Integration By Parts

: - \frac{x ^{2} - 25}{2 x ^{2}} - \int - \frac{1}{2 x x ^{2} - 25} d x

= - \frac{x ^{2} - 25}{2 x ^{2}} - \int - \frac{1}{2 x x ^{2} - 25} d x

\int - \frac{1}{2 x x ^{2} - 25} d x = - \frac{1}{10} arcsec (\frac{1}{5} x)

= - \frac{x ^{2} - 25}{2 x ^{2}} - (- \frac{1}{10} arcsec (\frac{1}{5} x))

Simplify

= - \frac{x ^{2} - 25}{2 x ^{2}} + \frac{1}{10} arcsec (\frac{1}{5} x)

Add a constant to the solution

= - \frac{x ^{2} - 25}{2 x ^{2}} + \frac{1}{10} arcsec (\frac{1}{5} x) + C

t^{2} y^{^{''}} + 15 t y^{^{'}} + 49 y = 0

f (x) = cot (x) csc (x)

x \to \infty lim (8 ln (x))

6 x + 1

(x - 3)^{\frac{2}{3}}

What is the integral of (sqrt(x^2-25))/(x^3) ?
The integral of (sqrt(x^2-25))/(x^3) is -(sqrt(x^2-25))/(2x^2)+1/10 arcsec(1/5 x)+C