Question

Evaluate ; `int x^2/(x^6 - 4x^3 + 13)` dx

Answer 1

Let I = `int x^2/(x^6 - 4x^3 + 13)` dx

        = `int x^2/((x^3)^2 - 4x^3 + 13)` dx

Put x³ = t

3x²dx = dt
x²dx = `dt/3`

I = `int ((dt/3))/(t^2 - 4t + 13)`

= `1/3 int ((dt))/[(t^2 - 4t + 4) - 4 + 13]`

= `1/3 1/[(t - 2)^2 + 3^2) dt`

= `1/3 . 1/3 tan^-1 ((t -2)/3) + c`

`int 1/(x^2 + a^2) dx = 1/a tan ^-1 (x/a) + c`

I = `1/9 tan^-1 ((x^3 - 2)/3) + c`