Question

Identify each of the following integrals or expressions as one of the functions of this chapter. Check your work by evaluating both your answer and the original problem by computer. Be sure you understand your computer program's notation.$$\int_{0}^{\infty} x^{3} e^{-x} d x$$

Short answer

The integral evaluates to 6 using the Gamma function.

Step-by-step solution

Identify the integral type

Recognize the given integral as an integral of the form \( \int_{0}^{\infty} x^{n} e^{-x} dx \), which is a Gamma function.

Recall the Gamma function definition

The Gamma function is defined as \( \Gamma(n+1) = \int_{0}^{\infty} x^{n} e^{-x} dx \). In this exercise, \ n = 3 \.

Apply the Gamma function result

Using the property of the Gamma function, \( \Gamma(4) = 3! = 6 \).

Check the result with a computer

Verify by evaluating \( \int_{0}^{\infty} x^{3} e^{-x} dx \) on a computer, ensuring \( 6 \) is the outcome.

Key concepts

integral calculus

Integral calculus is a branch of mathematics focused on the concepts of integration and the techniques to calculate integrals. It aims to find quantities like areas, volumes, and sums of infinite series. When facing an integral, it is crucial to understand its form to apply the correct solving technique.

For example, in the given problem, the integral of \( \ \int_{0}^{\infty} x^{3} e^{-x} \, \, dx\ \)n, we identify that it matches a specific type of integral commonly associated with the Gamma function. This recognition helps streamline the process of solving the integral.

Integral calculus encompasses:

• Definite Integrals: Calculated over a specific interval like \( \int_{a}^{b} \: f(x) dx \) .
• Indefinite Integrals: These are antiderivatives of functions, represented as \( \int f(x) dx \).
  
• Applications: Used to find areas under curves, volumes, and in physics equations.
  

Recognizing the integral's form and understanding which techniques to use is key in calculus.

Gamma function

The Gamma function, denoted as \(\Gamma(n)\) , is a broad extension of the factorial function to real and complex numbers. It’s defined by the integral:

\(\Gamma(n) = \int_{0}^{\infty} x^{n-1} e^{-x} dx \).

The Gamma function has several key properties:

• Factorial Connection: For positive integers, \( \Gamma(n) = (n-1)! \). For example, \( \Gamma(4) = 3! = 6 \), as shown in the exercise.
  
• Recursive Property: \(\Gamma(n+1) = n\Gamma(n) \).
• Value at \Gamma(1/2): \(\Gamma(1/2) = \sqrt{\pi} \)
  

The Gamma function is invaluable in various fields such as probability and statistics, physics, and engineering as it extends the applicability of the factorial function.

Understanding the Gamma function can greatly simplify evaluating integrals of the form \( \int_{0}^{\infty} x^{n-1} e^{-x} dx \).

computational verification

Computational verification involves using software tools to validate mathematical results. For integrals, programs like MATLAB, Mathematica, or Python libraries (e.g., SciPy) can be used to check the evaluation.

Here’s how to verify the result of the integral using Python and SciPy:
```pythonimport scipy.integrate as integrateresult, _ = integrate.quad(lambda x: x**3 * np.exp(-x), 0, np.inf)print(result) # Should print 6.0```This script computes the integral \(\ \int_{0}^{\infty} x^{3} e^{-x} dx \) precisely, reflecting the theoretical result of \(6 \).

Key aspects of computational verification:

• Accuracy: Ensures results are correct, preventing manual calculation errors.
  
• Efficiency: Saves time, especially with complex integrals.
  
• Understanding: Reinforces comprehension by seeing theoretical results match computed outcomes.
  

Overall, computational verification is a powerful method to confirm and deepen understanding of mathematical results.

integration techniques

Solving integrals can involve various techniques depending on the integral's structure. Here are some fundamental integration techniques:

• Substitution: Changes the variable to simplify the integral. For example, using \(u \)= \( x+2 \).
• Parts: Applies to the product of functions; uses \( \int u dv = uv - \int v du \).
• Partial Fractions: Decomposes complex fractions before integrating.
  
• Gamma Function: Specifically for integrals of form \( \int_{0}^{\infty} x^{n-1} e^{-x} dx \).
  

In the given problem, the integral aligns with the Gamma function, which allows simplifying calculations significantly:
\( \int_{0}^{\infty} x^{3} e^{-x} dx = \Gamma(4) = 3! = 6 \)

Using the right technique ensures the problem is approached methodically, leading to accurate solutions.

Mastering various integration techniques allows versatility in handling a wide range of integral problems.