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

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

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

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

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

Solution steps

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

Apply Integration By Parts

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

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

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

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

Simplify

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

Add a constant to the solution

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

\int_{0}^{2 π} sin^{2} (8 x) d x

\frac{d}{d x} (10 x ln (100 x))

\frac{\partial}{\partial y} (27 x y^{2} + 8 y^{3})

x \to 2 lim (4 x - 5)

\int \frac{1 - x + 2 x ^{2} - x ^{3}}{x ^{5} + 2 x ^{3} + x} d x

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