Question

Evaluate the following:

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

Answer 1

Let I = `int sqrt(1 + x^2)/x^4 "d"x`

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

= `int (xsqrt(1 + 1/x^2))/x^4 "d"x`

= `int (xsqrt(1 + 1/x^2))/x^4 "d"x`

Put `1 + 1/x^2` = r²

⇒ `(-2)/x^3 "d"x = 2"t" "dt"`

⇒ `- "dx"/x^3` = t dt

∴ I = `- int "t"^2 "dt"`

= `- "t"^3/3 + "C"`

= `- 1/3(1 + 1/x^2)^(3/2) + "C"`