Question

`int sqrt(x^2 - a^2)/x dx` = ______.

Options

• `sqrt(x^2 - a^2) - acos^-1(a/x) + c`

• `xsqrt(x^2 - a^2) - 1/atan^-1(x/a) + c`

• `sqrt(x^2 - a^2) + asec^-1(x/a) + c`

• `sqrt(x^2 - a^2) + 1/x sec^-1 (x) + c`

Answer 1

`int sqrt(x^2 - a^2)/x dx` = `underlinebb(sqrt(x^2 - a^2) - a  cos^-1(a/x) + c)`.

Explanation:

Let I = `int sqrt(x^2 - a^2)/x dx`

Put x = a secθ

`\implies` dx = a secθ tanθ dθ

`\implies I = int sqrt(a^2(sec^2 θ - 1))/(a sec θ). a sec θ tan θ  dθ` 

= `int a tan^2θ.dθ` = `aint(sec^2θ - 1)dθ`

= a (tanθ – θ) + c  ...`(∵ int sec^2x  dx = tanx)`

= `asqrt(sec^2θ - a) - a  θ + c`

= `asqrt((x^2/a^2) - a) - a sec^-1(x/a) + c` ...`(∵ sec θ = x/a)`

= `sqrt(x^2 - a^2) - a  sec^-1(x/a) + c`.

= `sqrt(x^2 - a^2) - a  cos^-1(a/x) + c`.