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

How to evaluate

\int\limits_{0}^{1} \mathrm{e}^{-x^{4}} (1-x^{4}) dx

was a question posed at Mathematics Stack Exchange. Here is my solution.

Let $$y=x^{4}$$

\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)

Using the same substitution, we also have

\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)

Thus we obtain

\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

We have used the lower incomplete gamma function:

\gamma(s,z) = \int\limits_{0}^{z} \mathrm{e}^{-x} x^{s-1} dx