Question

Integrate the rational function:

`1/(x(x^n + 1))` [Hint: multiply numerator and denominator by x^{n − 1} and put xⁿ = t]

Answer 1

Let `I = int 1/(x (x^n + 1))` dx

`= int x^(n - 1)/(x^n (x^n + 1))` dx

Put xⁿ = t

⇒ nx^{n -1} dx = dt

`therefore I = 1/n dt/(t (t + 1))`

Now, `1/(t(t + 1)) = A/t + B/(t + 1)`

∴ 1 = A(t + 1) + Bt

Putting t = 0, 1 = A

∴ A = 1

Putting t = -1, 1 = -1B

∴ B = -1

`therefore 1/(t(t + 1)) = 1/t - 1/(t + 1)`

`therefore I = 1/n  int dt/(t(t + 1)) = 1/n  int (1/t - 1/(t + 1))` dt

`= 1/n  log t - 1/n  log (t + 1) + C`

`= 1/n  [log t - log (t + 1)] + C`

`= 1/n  log abs (t/(t + 1)) + C`

`= 1/n   log abs ((x_n )/(x^n  + 1)) = C`