Question

Identify each of the following integrals or expressions as one of the functions of this chapter and then evaluate it using appropriate formulas and/or tables. $$ \int_{0}^{\infty} x^{3} e^{-x} d x $$

Short answer

6

Step-by-step solution

Identify the Integral

The given integral is of the form \( \int_{0}^{\infty} x^{n} e^{-x} d x \). This is a Gamma function.

Recall Gamma Function Definition

The Gamma function is defined as \( \Gamma(n+1) = \int_{0}^{\infty} x^{n} e^{-x} d x \).

Apply the Function

In our case, \( n = 3 \) . Therefore, we can write \( \Gamma(3+1) = \int_{0}^{\infty} x^{3} e^{-x} d x \).

Evaluate the Gamma Function

The Gamma function \( \Gamma(n+1) \) for integers is \( n! \). So, \( \Gamma(4) = 3! \).

Calculate the Factorial

The factorial of 3 is \( 3! = 3 \cdot 2 \cdot 1 = 6 \).

Conclusion

Therefore, \( \int_{0}^{\infty} x^{3} e^{-x} d x = 6 \).

Key concepts

Gamma Function

The Gamma function, represented as \(\text{\Gamma}(n)\), generalizes the concept of factorials to continuous values. It is defined as \(\text{\Gamma}(n) = \int_{0}^{\text{\infty}} x^{n-1} e^{-x} dx\). For positive integers, the Gamma function relates directly to factorials: \(\text{\Gamma}(n) = (n-1)!\). This means \text{\Gamma}(4) is equivalent to 3! = 6\.
The Gamma function has a unique property where \(\text{\Gamma}(n+1) = n\text{\Gamma}(n)\). This repetitive pattern simplifies integration involving exponential and power functions.

Definite Integral

A definite integral represents the area under a curve for a specific range. In the problem, the definite integral is \(\text{\int}_{0}^{\text{\infty}} x^3 e^{-x} dx\). This integral can be directly associated with the Gamma function.
We identified the integral by recognizing its form: \(\text{\int}_{0}^{\text{\infty}} x^n e^{-x} dx\) matches \(\text{\Gamma}(n+1)\). This shows how definite integrals can be evaluated using established formulas when they fit known functions.

Factorial

Factorial, denoted as \(!\), is the product of all positive integers up to a certain number. For example, \(3! = 3\cdot2\cdot1 = 6\). Factoring 6 in the exercise, \(\text{\Gamma}(4) = 3! \).
In general, if you have an integral that matches the Gamma function, you can convert it into a factorial solution. This shortcut makes complex integrals simpler by reducing them to basic arithmetic.

Exponential Functions

Exponential functions, represented as \(e^x\), are functions in which the variable is in the exponent. In our integral, the term \(e^{-x}\) denotes exponential decay.
When integrated with a polynomial function, like \(x^3 e^{-x}\), the integral simplifies to forms linked to known functions like the Gamma function.
The exponential function's unique properties, such as its derivative being itself, make it crucial in solving many calculus problems.