The integral we need to solve is:

\[
I = \int_0^{\infty} \frac{e^{4x}}{1 + e^{8x}} \, dx
\]

We can approach this using a substitution to simplify the integrand. Let’s proceed with the substitution \( u = e^{8x} \), which will lead to significant simplifications.

### Step 1: Substitution
Let \( u = e^{8x} \), so that:

\[
du = 8e^{8x}dx \quad \text{or} \quad dx = \frac{du}{8u}
\]

Also, when \( x = 0 \), \( u = 1 \), and when \( x \to \infty \), \( u \to \infty \). Thus, the limits of integration change accordingly:

\[
I = \int_1^{\infty} \frac{u^{1/2}}{(1 + u)} \cdot \frac{du}{8u}
\]

This simplifies to:

\[
I = \frac{1}{8} \int_1^{\infty} \frac{u^{1/2}}{u(1 + u)} \, du = \frac{1}{8} \int_1^{\infty} \frac{1}{u^{1/2}(1 + u)} \, du
\]

### Step 2: Break down the integral
Now, rewrite the integrand as:

\[
I = \frac{1}{8} \int_1^{\infty} \frac{1}{\sqrt{u}(1 + u)} \, du
\]

This integral is a standard form and can be solved using advanced techniques, leading to the following result:

\[
I = \frac{\pi}{8}
\]

Thus, the value of the integral is:

\[
\boxed{\frac{\pi}{8}}
\]

Would you like more details on the steps? Here are some follow-up questions for deeper exploration:

1. How does the substitution method work for simplifying integrals?
2. Can you explain other integral techniques for complex rational functions?
3. What are some common strategies for handling integrals with exponential functions?
4. What are special functions, and how do they relate to definite integrals like this?
5. How is the gamma function involved in solving such integrals?

**Tip:** When solving integrals, always check if a substitution or a transformation can simplify the integrand!

integral(0 to infinity) e^(4x) / (1 + e^(8x))

Explore how to solve the definite integral ∫_0^∞ e^(4x) / (1 + e^(8x)) dx using substitution. This solution involves transforming the integral and using advanced calculus techniques, resulting in a final answer of π/8.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation