Certainly! Here is the transcription formatted for an educational website:---### Calculus Integral Problem**Problem 81:**Evaluate the following integral:\[\int \frac{e^{3x} + e^{2x} + e^x}{(e^{2x} + 1)^2} \, dx\]**Explanation:**In this problem, we are asked to find the indefinite integral of a rational function with an exponential numerator and a polynomial denominator.---Use this transcription to understand and solve the integral problem. If needed, review techniques for integrating rational functions and handling exponential expressions. **76-83. Preliminary Steps**The following integrals require a preliminary step such as a change of variables before using the method of partial fractions. Evaluate these integrals.76. \[ \int \frac{\cos \theta}{d\theta} \]

The integration of (e3x + e2x +ex)/(e2x+1)2

Answered: (e3 + e2 + e) dx 81. (e20 + 1)? dr

Math

Calculus

(e3 + e2 + e) dx 81. (e20 + 1)? dr

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter1: Fundamental Concepts Of Algebra

Section1.2: Exponents And Radicals

Problem 11E

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Certainly! Here is the transcription formatted for an educational website:

---

### Calculus Integral Problem

**Problem 81:**

Evaluate the following integral:

\[
\int \frac{e^{3x} + e^{2x} + e^x}{(e^{2x} + 1)^2} \, dx
\]

**Explanation:**

In this problem, we are asked to find the indefinite integral of a rational function with an exponential numerator and a polynomial denominator.

---

Use this transcription to understand and solve the integral problem. If needed, review techniques for integrating rational functions and handling exponential expressions.

**76-83. Preliminary Steps**

The following integrals require a preliminary step such as a change of variables before using the method of partial fractions. Evaluate these integrals.

76.
\[ \int \frac{\cos \theta}{d\theta} \]

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