What is the integral of (sqrt(x^2+4))/(x^2) ?

The integral of (sqrt(x^2+4))/(x^2) is -(sqrt(x^2+4))/x+ln(1/2 |x+sqrt(4+x^2)|)+C

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integral of (sqrt(x^2+4))/(x^2)

\int \frac{x ^{2} + 4}{x ^{2}} d x

- \frac{x ^{2} + 4}{x} + ln (\frac{1}{2} x + 4 + x^{2}) + C

Solution steps

\int \frac{x ^{2} + 4}{x ^{2}} d x

Apply Integration By Parts

: - \frac{x ^{2} + 4}{x} - \int - \frac{1}{x ^{2} + 4} d x

= - \frac{x ^{2} + 4}{x} - \int - \frac{1}{x ^{2} + 4} d x

\int - \frac{1}{x ^{2} + 4} d x = - ln (\frac{1}{2} x + 4 + x^{2})

= - \frac{x ^{2} + 4}{x} - (- ln (\frac{1}{2} x + 4 + x^{2}))

Simplify

= - \frac{x ^{2} + 4}{x} + ln (\frac{1}{2} x + 4 + x^{2})

Add a constant to the solution

= - \frac{x ^{2} + 4}{x} + ln (\frac{1}{2} x + 4 + x^{2}) + C

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What is the integral of (sqrt(x^2+4))/(x^2) ?
The integral of (sqrt(x^2+4))/(x^2) is -(sqrt(x^2+4))/x+ln(1/2 |x+sqrt(4+x^2)|)+C