Let's do some calculus! (6)

Calculus Level 4

\[\begin{align} I &= \int{\dfrac{e^{x}}{e^{4x}+e^{2x}+1}}\,dx \\ J & = \int{\dfrac{e^{-x}}{e^{-4x}+e^{-2x}+1}}\,dx \end{align} \]

For \(I\) and \(J\) as defined above, find \(J-I\).

Notations:

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

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