Calculus: Early Transcendentals
8th Edition
ISBN: 9781285741550
Author: James Stewart
Publisher: Cengage Learning
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![Title: Finding Indefinite Integrals
---
In this tutorial, we will learn how to find indefinite integrals of a given function. Consider the following expression:
\[ \int (3x^4 + 4)^5 \cdot 12x^3 \, dx \]
We aim to integrate this function with respect to \(x\).
**Explanation:**
1. Identify the integral expression.
2. Recognize if a substitution method can simplify the integration.
**Step-by-Step Integration:**
1. **Substitution Method:**
- Let \( u = 3x^4 + 4 \).
- Thus, the differential \( du \) can be found by differentiating \( u \) with respect to \( x \):
\[
\frac{du}{dx} = 12x^3
\]
or equivalently,
\[
du = 12x^3 \, dx
\]
2. **Rewrite the Integral:**
- Substitute \( u \) and \( du \) back into the integral expression:
\[
\int (3x^4 + 4)^5 \cdot 12x^3 \, dx = \int u^5 \, du
\]
3. **Integrate:**
- Integrate \( u^5 \) with respect to \( u \):
\[
\int u^5 \, du = \frac{u^6}{6} + C
\]
where \( C \) is the constant of integration.
4. **Back-Substitute \( u \):**
- Replace \( u \) with the original expression \( 3x^4 + 4 \):
\[
\frac{(3x^4 + 4)^6}{6} + C
\]
5. **Final Answer:**
\[
\int (3x^4 + 4)^5 \cdot 12x^3 \, dx = \frac{(3x^4 + 4)^6}{6} + C
\]
By following these steps, you can solve similar indefinite integral problems using substitution and integration techniques.](https://content.bartleby.com/qna-images/question/239df334-5b03-4bc0-9001-d398335a6cd3/bf506386-e6d0-4a7d-b91e-4197bf604654/bx5ktub_reoriented.jpeg)
Transcribed Image Text:Title: Finding Indefinite Integrals
---
In this tutorial, we will learn how to find indefinite integrals of a given function. Consider the following expression:
\[ \int (3x^4 + 4)^5 \cdot 12x^3 \, dx \]
We aim to integrate this function with respect to \(x\).
**Explanation:**
1. Identify the integral expression.
2. Recognize if a substitution method can simplify the integration.
**Step-by-Step Integration:**
1. **Substitution Method:**
- Let \( u = 3x^4 + 4 \).
- Thus, the differential \( du \) can be found by differentiating \( u \) with respect to \( x \):
\[
\frac{du}{dx} = 12x^3
\]
or equivalently,
\[
du = 12x^3 \, dx
\]
2. **Rewrite the Integral:**
- Substitute \( u \) and \( du \) back into the integral expression:
\[
\int (3x^4 + 4)^5 \cdot 12x^3 \, dx = \int u^5 \, du
\]
3. **Integrate:**
- Integrate \( u^5 \) with respect to \( u \):
\[
\int u^5 \, du = \frac{u^6}{6} + C
\]
where \( C \) is the constant of integration.
4. **Back-Substitute \( u \):**
- Replace \( u \) with the original expression \( 3x^4 + 4 \):
\[
\frac{(3x^4 + 4)^6}{6} + C
\]
5. **Final Answer:**
\[
\int (3x^4 + 4)^5 \cdot 12x^3 \, dx = \frac{(3x^4 + 4)^6}{6} + C
\]
By following these steps, you can solve similar indefinite integral problems using substitution and integration techniques.
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