Question

Find:

`int1/x sqrt((x + a)/(x - a))  dx`

Answer 1

`int1/x sqrt((x + a)/(x - a))  dx`

`int1/x sqrt((x + a)/(x - a) xx (x - a)/(x - a))  dx`

`intsqrt((x^2 - a^2))/(x(x - a))  dx`

Let x = a sec θ

dx = a sec θ tan θ dθ

= `int(sqrt(a^2(sec^2 θ - 1)) xx a sec θ tan θ  dθ)/(a sec θ xx a(sec θ- 1))`

= `int(a tan θ xx a sec θ tan θ  dθ)/(a xx a sec θ(sec θ - 1))`

= `int (tan^2 θ  dθ)/(sec θ - 1)`

= `int(sec^2 θ - 1^2)/(sec θ - 1)  dθ`

= `int((sec θ - 1)(sec θ + 1))/(sec θ - 1)  dθ`

= `int(sec θ + 1) dθ`

= log |sec θ + tan θ| + θ + c    ...`[{:(∵ x = a sec θ","  θ = sec^-1  x/a),(sec θ = x/a"," tan θ = sqrt(x^2/a^2 - 1)):}]`

I = `log|x/a + sqrt(x^2/a^2 - 1)| + sec^-1  x/a + c_1`

= `log|(x + sqrt(x^2 - a^2))/a| + sec^-1  x/a + c_1`

= `log |x + sqrt(x^2 - a^2)| + sec^-1  x/a + c_1 - loga`

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

Where C = C₁ − log a