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

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

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

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

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

Solution steps

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

Apply Integration By Parts

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

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

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

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

Simplify

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

Add a constant to the solution

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

\int \frac{x ^{3} + x}{( x ^{2} + 3 ) ( x ^{2} - 2 x + 3 )} d x

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

x \to 0 lim (\frac{x - sin ( x )}{x})

\frac{\partial}{\partial y} (y sin (y))

y = 8 x sin (x^{2})

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