What is the integral of (sqrt(9x^2-16))/(x^3) ?

The integral of (sqrt(9x^2-16))/(x^3) is -(sqrt(9x^2-16))/(2x^2)+9/8 arcsec(3/4 x)+C

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integral of (sqrt(9x^2-16))/(x^3)

\int \frac{9 x ^{2} - 16}{x ^{3}} d x

- \frac{9 x ^{2} - 16}{2 x ^{2}} + \frac{9}{8} arcsec (\frac{3}{4} x) + C

Solution steps

\int \frac{9 x ^{2} - 16}{x ^{3}} d x

Apply Integration By Parts

: - \frac{9 x ^{2} - 16}{2 x ^{2}} - \int - \frac{9}{2 x 9 x ^{2} - 16} d x

= - \frac{9 x ^{2} - 16}{2 x ^{2}} - \int - \frac{9}{2 x 9 x ^{2} - 16} d x

\int - \frac{9}{2 x 9 x ^{2} - 16} d x = - \frac{9}{8} arcsec (\frac{3}{4} x)

= - \frac{9 x ^{2} - 16}{2 x ^{2}} - (- \frac{9}{8} arcsec (\frac{3}{4} x))

Simplify

= - \frac{9 x ^{2} - 16}{2 x ^{2}} + \frac{9}{8} arcsec (\frac{3}{4} x)

Add a constant to the solution

= - \frac{9 x ^{2} - 16}{2 x ^{2}} + \frac{9}{8} arcsec (\frac{3}{4} x) + C

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What is the integral of (sqrt(9x^2-16))/(x^3) ?
The integral of (sqrt(9x^2-16))/(x^3) is -(sqrt(9x^2-16))/(2x^2)+9/8 arcsec(3/4 x)+C