To solve the integral [tex]\(\int 3 x^9 e^{x^3} \, dx\)[/tex], we'll perform the following steps:
1. Identify the substitution:
Let's set a substitution to simplify the integral. Notice the exponent in the integrand is [tex]\(x^3\)[/tex]. This suggests that setting [tex]\(u = x^3\)[/tex] could simplify the integral.
2. Compute the differential [tex]\(du\)[/tex]:
If [tex]\(u = x^3\)[/tex], then the differential [tex]\(du\)[/tex] is computed as follows:
[tex]\[
u = x^3 \implies du = 3x^2 \, dx \implies dx = \frac{du}{3x^2}
\][/tex]
3. Rewrite the integral in terms of [tex]\(u\)[/tex]:
Substitute [tex]\(u = x^3\)[/tex] into the integral and note that [tex]\(x^9 = (x^3)^3 = u^3\)[/tex]:
[tex]\[
\int 3 x^9 e^{x^3} \, dx = \int 3 u^3 e^u \cdot \frac{du}{3x^2}
\][/tex]
Since [tex]\(x^2 = \left( u^{1/3} \right)^2 = u^{2/3}\)[/tex], then:
[tex]\[
dx = \frac{du}{3 u^{2/3}}
\][/tex]
Thus, the integral becomes:
[tex]\[
\int 3 u^3 e^u \cdot \frac{du}{3 u^{2/3}} = \int u^3 e^u \cdot \frac{du}{u^{2/3}}
\][/tex]
Simplify the exponents:
[tex]\[
\int u^3 e^u \cdot u^{-2/3} \, du = \int u^{3-(2/3)} e^u \, du = \int u^{7/3} e^u \, du
\][/tex]
4. Use integration by parts or special functions:
The integral [tex]\(\int u^{7/3} e^u \, du\)[/tex] does not easily simplify using basic integration techniques. It involves special functions known as the incomplete gamma functions or lower gamma functions.
5. Express the integral in terms of special functions:
This integral can be expressed using the lower gamma function, [tex]\(\gamma(s, x)\)[/tex], defined as:
[tex]\[
\gamma(s, x) = \int_0^x t^{s-1} e^{-t} dt
\][/tex]
The lower gamma function provides a way to represent integrals of the form [tex]\(\int u^n e^u \, du\)[/tex].
6. Solve using the gamma and lower gamma functions:
For our integral, we get:
[tex]\[
\int u^{7/3} e^u \, du = \frac{10 \cdot \exp\left(\frac{2i\pi}{3}\right) \cdot \Gamma\left(\frac{10}{3}\right) \cdot \gamma\left(\frac{10}{3}, u \cdot \exp(\pi i)\right)}{3 \cdot \Gamma\left(\frac{13}{3}\right)}
\][/tex]
Finally, substitute [tex]\(u = x^3\)[/tex] back into the expression:
[tex]\[
\frac{10 \cdot \exp\left(\frac{2i\pi}{3}\right) \cdot \Gamma\left(\frac{10}{3}\right) \cdot \gamma\left(\frac{10}{3}, x^3 \cdot \exp(\pi i)\right)}{3 \cdot \Gamma\left(\frac{13}{3}\right)}
\][/tex]
So, the integral [tex]\(\int 3 x^9 e^{x^3} \, dx\)[/tex] evaluates to:
[tex]\[
\frac{10 \cdot \exp\left(\frac{2i\pi}{3}\right) \cdot \Gamma\left(\frac{10}{3}\right) \cdot \gamma\left(\frac{10}{3}, x^3 \cdot \exp(\pi i)\right)}{3 \cdot \Gamma\left(\frac{13}{3}\right)}
\][/tex]
