https://file.aiquickdraw.com/mathbot/file/1727103952967a9cl5xnz.jpg

The integral we are tasked with solving is:

\[
\int \frac{e^x}{1 + e^{2x}} \, dx
\]

### Step-by-Step Solution:

#### 1. **Substitution**:
   Let's try a substitution to simplify the expression. Let:
   \[
   u = e^x
   \]
   Then, the derivative of \( u \) with respect to \( x \) is:
   \[
   du = e^x \, dx
   \]
   This transforms the integral into:
   \[
   \int \frac{du}{1 + u^2}
   \]

#### 2. **Recognizing a Standard Integral**:
   The integral \( \int \frac{du}{1 + u^2} \) is a standard form, and its result is:
   \[
   \int \frac{du}{1 + u^2} = \tan^{-1}(u)
   \]

#### 3. **Substitute Back**:
   Now, we substitute back \( u = e^x \):
   \[
   \int \frac{e^x}{1 + e^{2x}} \, dx = \tan^{-1}(e^x) + C
   \]

### Final Answer:
\[
\int \frac{e^x}{1 + e^{2x}} \, dx = \tan^{-1}(e^x) + C
\]
where \( C \) is the constant of integration.

---

Do you want any further explanation or details about this solution?

Here are five follow-up questions to deepen understanding:
1. What is the purpose of using substitution in integrals?
2. How do you identify when to use the substitution method?
3. Why is the integral \( \frac{1}{1 + u^2} \) linked to \( \tan^{-1}(u) \)?
4. What are some other integrals that result in inverse trigonometric functions?
5. How would the result change if the denominator were \( 1 + e^x \) instead of \( 1 + e^{2x} \)?

**Tip**: Always check if the integrand can be simplified using a substitution, as it often turns a complex expression into a recognizable form.

Evaluate the integral \(\int \frac{e^x}{1 + e^{2x}} dx\). Use a capital C to denote an arbitrary constant of integration.

This page walks through how to evaluate the integral \(\int \frac{e^x}{1 + e^{2x}} dx\) using substitution. Learn how to simplify the expression and apply the standard arctangent integral formula. Ideal for students studying Calculus 2.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation