The integral is equal to: #01

The integral \( \int \dfrac{2x^{12}+5x^9}{(x^5+x^3+1)^3} dx \) is equal to:

  1. \( \dfrac{-x^5}{(x^5+x^3+1)^2} + C \)
  2. \( \dfrac{x^{10}}{2(x^5+x^3+1)^2} + C \)
  3. \( \dfrac{x^5}{2(x^5+x^3+1)^2} + C \)
  4. \( \dfrac{-x^{10}}{2(x^5+x^3+1)^2} + C \)

Solution

\( \int \dfrac{2x^{12}+5x^9}{(x^5+x^3+1)^3} dx \)

Divide numerator and denominator by x15

\( \int \dfrac{2/x^3+5/x^6}{(1+1/x^2+1/x^5)^3} dx \)

Let \( t = 1+ \dfrac{1}{x^2} + \dfrac{1}{x^5} \)

\( dt = (- \dfrac{2}{x^3} - \dfrac{5}{x^6}) dx \)

The integral changes to:

\( \int \dfrac{-dt}{t^3} = \dfrac{1}{2t^2} \)

Put the value of t

\( I = \dfrac{x^{10}}{2(x^5+x^3+1)^2} + c \)

The correct option is B.