Find the indefinite integral of e^(-x) times x^3.102378 with respect to x.

Step-by-step solution

1. 1

   See why integration by parts fails here. For an integer power $x^n e^{-x}$, repeated by-parts lowers the exponent one step at a time until it reaches $x^0$ and the integral closes. With $a = 3.102378$ the exponents run $3.102378,\ 2.102378,\ 1.102378,\ 0.102378,\ -0.897622,\dots$ and never hit zero, so the process never terminates. No elementary antiderivative exists.

2. 2

   Name the special function that does close it. The lower incomplete gamma function is defined as

   $\gamma(s,x) = \int_0^{x} t^{s-1}e^{-t}\,dt$

   By the Fundamental Theorem of Calculus, $\dfrac{d}{dx}\gamma(s,x) = x^{s-1}e^{-x}$.

3. 3

   Match the exponent. We need $s - 1 = 3.102378$, so

   $s = 4.102378$

   and therefore

   $\int e^{-x}x^{3.102378}\,dx = \gamma(4.102378,\,x) + C$

4. 4

   State the equivalent upper-gamma form. Since $\gamma(s,x) + \Gamma(s,x) = \Gamma(s)$ with $\Gamma(s)$ constant in $x$, the same antiderivative can be written

   $\int e^{-x}x^{3.102378}\,dx = -\Gamma(4.102378,\,x) + C$

   The two forms differ only by the constant $\Gamma(4.102378)$, which is absorbed into $C$.

5. 5

   Read off the definite integral over the whole ray. Letting $x \to \infty$ collapses the incomplete gamma to the complete one:

   $\int_0^{\infty} e^{-x}x^{3.102378}\,dx = \Gamma(4.102378) \approx 6.83346$

   For comparison, the neighbouring integer case is $\Gamma(4) = 3! = 6$, so the slightly larger exponent lifts the value modestly, as expected from the smooth growth of $\Gamma$ past $s = 4$.