Let's do some calculus! (6)

Calculus Level 5

I=exe4x+e2x+1dxJ=exe4x+e2x+1dx\begin{aligned} I &= \int{\dfrac{e^{x}}{e^{4x}+e^{2x}+1}}\,dx \\ J & = \int{\dfrac{e^{-x}}{e^{-4x}+e^{-2x}+1}}\,dx \end{aligned}

For II and JJ as defined above, find JIJ-I.

Notations:

  • e2.718e \approx 2.718 is the Euler's number.
  • log()\log (\cdot) denotes the natural logarithm function, that is loge()\log_{e}{(\cdot)} or ln()\ln(\cdot).
  • CC denotes the constant of integration.
  • xR+x \in \mathbb{R}^{+}

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