What is the integral of (sqrt(25x^2-64))/(x^3) ?

The integral of (sqrt(25x^2-64))/(x^3) is -(sqrt(25x^2-64))/(2x^2)+25/16 arcsec(5/8 x)+C

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integral of (sqrt(25x^2-64))/(x^3)

\int \frac{25 x ^{2} - 64}{x ^{3}} d x

- \frac{25 x ^{2} - 64}{2 x ^{2}} + \frac{25}{16} arcsec (\frac{5}{8} x) + C

Solution steps

\int \frac{25 x ^{2} - 64}{x ^{3}} d x

Apply Integration By Parts

: - \frac{25 x ^{2} - 64}{2 x ^{2}} - \int - \frac{25}{2 x 25 x ^{2} - 64} d x

= - \frac{25 x ^{2} - 64}{2 x ^{2}} - \int - \frac{25}{2 x 25 x ^{2} - 64} d x

\int - \frac{25}{2 x 25 x ^{2} - 64} d x = - \frac{25}{16} arcsec (\frac{5}{8} x)

= - \frac{25 x ^{2} - 64}{2 x ^{2}} - (- \frac{25}{16} arcsec (\frac{5}{8} x))

Simplify

= - \frac{25 x ^{2} - 64}{2 x ^{2}} + \frac{25}{16} arcsec (\frac{5}{8} x)

Add a constant to the solution

= - \frac{25 x ^{2} - 64}{2 x ^{2}} + \frac{25}{16} arcsec (\frac{5}{8} x) + C

\int 4 ln (t) d t

g (x) = sin (9 - 3 x)

f (x) = \frac{x}{9 x ^{2} - 1}

\frac{d}{d x} (ln (\frac{x + 3}{x - 3}))

x \to 0 + lim (\frac{x}{15 x ^{4} + 12 x})

What is the integral of (sqrt(25x^2-64))/(x^3) ?
The integral of (sqrt(25x^2-64))/(x^3) is -(sqrt(25x^2-64))/(2x^2)+25/16 arcsec(5/8 x)+C