https://file.aiquickdraw.com/mathbot/file/1723182025089gfkmoql3.jpg

The integral given in the image is:

\[
\int_0^\infty x^7 e^{-2x} \, dx
\]

This is a standard form that can be solved using the Gamma function. The Gamma function is defined as:

\[
\Gamma(n) = \int_0^\infty x^{n-1} e^{-x} \, dx
\]

For this specific integral, we can rewrite it in the form of the Gamma function by factoring out constants and making a substitution:

### Step 1: Identify the structure
The integral is:

\[
\int_0^\infty x^7 e^{-2x} \, dx
\]

### Step 2: Make a substitution to match the Gamma function form
Let \( u = 2x \), hence \( du = 2dx \) or \( dx = \frac{du}{2} \).

### Step 3: Substitute into the integral
Substituting \( u = 2x \) into the integral gives:

\[
\int_0^\infty x^7 e^{-2x} \, dx = \int_0^\infty \left(\frac{u}{2}\right)^7 e^{-u} \cdot \frac{du}{2}
\]

Simplify this:

\[
\int_0^\infty \frac{u^7}{2^7} e^{-u} \cdot \frac{du}{2} = \frac{1}{2^8} \int_0^\infty u^7 e^{-u} \, du
\]

### Step 4: Recognize the Gamma function
The integral \( \int_0^\infty u^7 e^{-u} \, du \) is the Gamma function \( \Gamma(8) \).

\[
\Gamma(8) = (7)!
\]

So, the integral becomes:

\[
\frac{1}{2^8} \cdot \Gamma(8) = \frac{7!}{2^8}
\]

### Step 5: Calculate the factorial and power of 2
\[
7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 5040
\]

\[
2^8 = 256
\]

### Final Answer:
\[
\int_0^\infty x^7 e^{-2x} \, dx = \frac{5040}{256} = 19.6875
\]

So, the value of the integral is **19.6875**.

Would you like to explore any details or have any questions?

Here are some related questions:
1. What is the general formula for the Gamma function and its properties?
2. How do you derive the Gamma function for non-integer values?
3. What are the applications of the Gamma function in different areas of mathematics?
4. Can we solve integrals of different forms using the Gamma function?
5. How does the substitution method help in solving integrals?
6. What is the relationship between the Gamma function and factorials?
7. Can you provide examples where the Gamma function is used in physics or engineering?
8. How do other special functions relate to the Gamma function?

**Tip:** The Gamma function is a powerful tool for evaluating integrals that involve factorial-like expressions, and it generalizes the concept of factorials to non-integer values.

Learn how to solve the integral ∫₀ᴵⁿᶠ x⁷ e⁻²ᵘ dx using the Gamma function and substitution method. This detailed guide covers the Gamma function definition, substitution method for integrals, and step-by-step solution for the specific integral. Suitable for advanced college-level students and beyond.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation