Evaluate the Integral \( \int_{0}^{1} \mathrm{e}^{-x^{4}} (1-x^{4}) dx \)

How to evaluate
\begin{equation}
\int\limits_{0}^{1} \mathrm{e}^{-x^{4}} (1-x^{4}) dx
\end{equation}
was a question posed at Mathematics Stack Exchange. Here is my solution.

Let \(y=x^{4}\)
\begin{equation}
\int\limits_{0}^{1} \mathrm{e}^{-x^{4}} dx = \frac{1}{4} \int\limits_{0}^{1} \mathrm{e}^{-y} y^{-3/4} dy
= \frac{1}{4} \gamma\left(\frac{1}{4},1 \right)
\end{equation}

Using the same substitution, we also have
\begin{equation}
\int\limits_{0}^{1} \mathrm{e}^{-x^{4}} x^{4} dx = \frac{1}{4} \int\limits_{0}^{1} \mathrm{e}^{-y} y^{1/4} dy
= \frac{1}{4} \gamma\left(\frac{5}{4},1 \right)
\end{equation}

Thus we obtain
\begin{equation}
\int\limits_{0}^{1} \mathrm{e}^{-x^{4}} (1-x^{4}) dx
= \frac{1}{4} \Big[ \gamma\left(\frac{1}{4},1 \right) – \gamma\left(\frac{5}{4},1 \right) \Big]
\approx 0.7256
\end{equation}

We have used the lower incomplete gamma function:
\begin{equation}
\gamma(s,z) = \int\limits_{0}^{z} \mathrm{e}^{-x} x^{s-1} dx
\end{equation}

Leave a Reply

Your email address will not be published. Required fields are marked *