Question

Evaluate the following:

`int ("d"x)/(xsqrt(x^4 - 1))`  (Hint: Put x² = sec θ)

Answer 1

Let I = `int ("d"x)/(xsqrt(x^4 - 1))` 

= `int (x"d"x)/(x^2sqrt(x^4 - 1))`

Put x² = sec θ

∴ 2x dx = sec θ tan θ dθ

x dx = `1/2 sec theta tan theta  "d"theta`

∴ I = `1/2 int (sec theta tan theta)/(sec theta sqrt(sec^2theta - 1)) "d"theta`

= `1/2 int (sectheta tan theta)/(sectheta * tan theta) "d"theta`

= `1/2 int 1 "d"theta`

= `1/2 theta + "C"`

So I = `1/2 sec^-1x^2 + "C"`

Hence, I = `1/2 sec^-1 x^2 + "C"`.