What is the integral of (sqrt(x^2+1))/(x^2) ?

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

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integral of (sqrt(x^2+1))/(x^2)

\int \frac{x ^{2} + 1}{x ^{2}} d x

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

Solution steps

\int \frac{x ^{2} + 1}{x ^{2}} d x

Apply Integration By Parts

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

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

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

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

Simplify

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

Add a constant to the solution

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

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