The integral 2x^12+5x^9(x^5+x^3+1)^3,dx is equal to:

Question

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

Accepted Answer

Dividing numerator and denominator by $x^{15}$: $\int \dfrac{\left(\dfrac{2}{x^3}+\dfrac{5}{x^6}\right)dx}{\left(1+\dfrac{1}{x^2}+\dfrac{1}{x^5}\right)^3}$

Let $1+\dfrac{1}{x^2}+\dfrac{1}{x^5}=t \Rightarrow \left(-\dfrac{2}{x^3}-\dfrac{5}{x^6}\right)dx = dt$

$\int \dfrac{-dt}{t^3} = \dfrac{1}{2t^2}+C = \dfrac{1}{2\left(1+\dfrac{1}{x^2}+\dfrac{1}{x^5}\right)^2}+C = \dfrac{x^{10}}{2(x^5+x^3+1)^2}+C$

Solutions · 1