What is the integral of (sqrt(x^2-a^2))/(x^2) ?

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

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integral of (sqrt(x^2-a^2))/(x^2)

\int \frac{x ^{2} - a ^{2}}{x ^{2}} d x

- \frac{x ^{2} - a ^{2}}{x} + ln (\frac{x ^{2} - a ^{2} + x}{∣ a ∣}) + C

Solution steps

\int \frac{x ^{2} - a ^{2}}{x ^{2}} d x

Apply Integration By Parts

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

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

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

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

Simplify

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

Add a constant to the solution

= - \frac{x ^{2} - a ^{2}}{x} + ln (\frac{x ^{2} - a ^{2} + x}{∣ a ∣}) + C

\int \frac{1}{2} cos (x) d x

y \to 2 lim (\frac{y ^{3} - 6 y ^{2} + 12 y - 8}{y ^{4} - 4 y ^{3} + 16 y - 16})

\frac{d y}{d x} = x - x y

x \to 4 + lim (x^{2} - 16)

x \to 0 lim (\frac{1 ^{2 x} - 1}{x})

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