Question

Integrate the following with respect to x.

`1/(x^2(x^2 + 1))`

Answer 1

`1/(x^2(x^2 + 1)) = x/(x^2(x^2 + 1))`

Let x² = t

Then 2x dx = dt

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

= `1/2 int "dt"/("t"("t" + 1))`

= `1/2 int [1/"t" - 1/("t" + 1)]  "dt"`  ......(Using partial fractions)

= `1/2 [log |"t"| - log |"t" + 1|] + "c"`

= `1/2 log |"t"/("t" + 1)|`

Replacing t by x² we get = `1/2 log |x^2/(x^2 + 1)| + "c"`

= `log|x| - 1/2 log|x^2 + 1| + "c"`