If m is a non-zero number and x^5m-1+2x^4m-1(x^2m+x^m+1)^3,dx = f(x)+c, then f(x) is:

Question

If $m$ is a non-zero number and $\displaystyle\int \dfrac{x^{5m-1}+2x^{4m-1}}{(x^{2m}+x^m+1)^3}\,dx = f(x)+c$, then $f(x)$ is:

Accepted Answer

$\int \dfrac{x^{5m-1}+2x^{4m-1}}{(x^{2m}+x^m+1)^3}\,dx = \int \dfrac{x^{-m-1}+2x^{-2m-1}}{(1+x^{-m}+x^{-2m})^3}\,dx$

Put $t = 1+x^{-m}+x^{-2m}$: $\dfrac{dt}{dx} = -mx^{-m-1}-2mx^{-2m-1} \Rightarrow \dfrac{dt}{-m} = (x^{-m-1}+2x^{-2m-1})\,dx$

$\int \dfrac{x^{5m-1}+2x^{4m-1}}{(x^{2m}+x^m+1)^3}\,dx = \dfrac{1}{-m}\int t^{-3}\,dt = \dfrac{1}{2mt^2}+C$

$= \dfrac{1}{2m(1+x^{-m}+x^{-2m})^2}+C = \dfrac{x^{4m}}{2m(x^{2m}+x^m+1)^2}+C$

$	herefore f(x) = \dfrac{x^{4m}}{2m(x^{2m}+x^m+1)^2}$

Solutions · 1