2. f(e³x + √x +9)dx 3x

Calculus: Early Transcendentals
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Author:James Stewart
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Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Please show how to integrate by hand
Certainly! Here's the transcription and explanation for an educational website:

---

### Integral Problem

**Problem 2:**

Evaluate the integral:

\[ \int \left( e^{3x} + \sqrt[3]{x + 9} \right) dx \]

---

In this problem, you are required to solve the integral of a function that comprises two parts:

1. The exponential function \( e^{3x} \).
2. The cube root function \( \sqrt[3]{x + 9} \).

#### Steps to solve:

1. **Integral of \( e^{3x} \):**

   The integral of \( e^{kx} \) with respect to \( x \) is \( \frac{1}{k} e^{kx} + C \), where \( C \) is the constant of integration.

   For our function:
   \[ \int e^{3x} dx = \frac{1}{3} e^{3x} + C \]

2. **Integral of \( \sqrt[3]{x + 9} \):**

   Rewrite \( \sqrt[3]{x + 9} \) as \( (x + 9)^{\frac{1}{3}} \).

   The integral of \( (x + a)^n \) is \( \frac{(x + a)^{n+1}}{n+1} + C \), where \( C \) is the constant of integration.

   For our function:
   \[ \int (x + 9)^{\frac{1}{3}} dx = \frac{3}{4} (x + 9)^{\frac{4}{3}} + C \]

3. Combine the results:

   \[ \int \left( e^{3x} + \sqrt[3]{x + 9} \right) dx = \frac{1}{3} e^{3x} + \frac{3}{4} (x + 9)^{\frac{4}{3}} + C \]

This is the antiderivative of the given function. The constant of integration, \( C \), represents any constant value since the derivative of a constant is zero.

---

This solution process demonstrates the integration techniques for exponential and root functions, crucial for calculus students.
Transcribed Image Text:Certainly! Here's the transcription and explanation for an educational website: --- ### Integral Problem **Problem 2:** Evaluate the integral: \[ \int \left( e^{3x} + \sqrt[3]{x + 9} \right) dx \] --- In this problem, you are required to solve the integral of a function that comprises two parts: 1. The exponential function \( e^{3x} \). 2. The cube root function \( \sqrt[3]{x + 9} \). #### Steps to solve: 1. **Integral of \( e^{3x} \):** The integral of \( e^{kx} \) with respect to \( x \) is \( \frac{1}{k} e^{kx} + C \), where \( C \) is the constant of integration. For our function: \[ \int e^{3x} dx = \frac{1}{3} e^{3x} + C \] 2. **Integral of \( \sqrt[3]{x + 9} \):** Rewrite \( \sqrt[3]{x + 9} \) as \( (x + 9)^{\frac{1}{3}} \). The integral of \( (x + a)^n \) is \( \frac{(x + a)^{n+1}}{n+1} + C \), where \( C \) is the constant of integration. For our function: \[ \int (x + 9)^{\frac{1}{3}} dx = \frac{3}{4} (x + 9)^{\frac{4}{3}} + C \] 3. Combine the results: \[ \int \left( e^{3x} + \sqrt[3]{x + 9} \right) dx = \frac{1}{3} e^{3x} + \frac{3}{4} (x + 9)^{\frac{4}{3}} + C \] This is the antiderivative of the given function. The constant of integration, \( C \), represents any constant value since the derivative of a constant is zero. --- This solution process demonstrates the integration techniques for exponential and root functions, crucial for calculus students.
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