What is the integral of (sqrt(x^2+5))/(x^2) ?

The integral of (sqrt(x^2+5))/(x^2) is -(sqrt(x^2+5))/x+ln| 1/(sqrt(5))x+sqrt(1/5 (5+x^2))|+C

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integral of (sqrt(x^2+5))/(x^2)

\int \frac{x ^{2} + 5}{x ^{2}} d x

- \frac{x ^{2} + 5}{x} + ln \frac{1}{5} x + \frac{1}{5} (5 + x^{2}) + C

Solution steps

\int \frac{x ^{2} + 5}{x ^{2}} d x

Apply Integration By Parts

= - \frac{x ^{2} + 5}{x} - \int - \frac{1}{x ^{2} + 5} d x

\int - \frac{1}{x ^{2} + 5} d x = - ln \frac{1}{5} x + \frac{1}{5} (5 + x^{2})

= - \frac{x ^{2} + 5}{x} - (- ln \frac{1}{5} x + \frac{1}{5} (5 + x^{2}))

Simplify

- \frac{x ^{2} + 5}{x} - (- ln \frac{1}{5} x + \frac{1}{5} (5 + x^{2})) : - \frac{x ^{2} + 5}{x} + ln \frac{1}{5} x + \frac{1}{5} (5 + x^{2})

= - \frac{x ^{2} + 5}{x} + ln \frac{1}{5} x + \frac{1}{5} (5 + x^{2})

Add a constant to the solution

= - \frac{x ^{2} + 5}{x} + ln \frac{1}{5} x + \frac{1}{5} (5 + x^{2}) + C

What is the integral of (sqrt(x^2+5))/(x^2) ?
The integral of (sqrt(x^2+5))/(x^2) is -(sqrt(x^2+5))/x+ln| 1/(sqrt(5))x+sqrt(1/5 (5+x^2))|+C