Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
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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Question
![**Problem:**
Find the indefinite integral and check the results by differentiation.
\[
\int (4x^2 + 3)^2 \, dx
\]
**Solution:**
To solve this problem, we need to follow these steps:
1. **Expand the integrand:** First, expand \((4x^2 + 3)^2\).
Using the formula \((a + b)^2 = a^2 + 2ab + b^2\), we get:
\[
(4x^2 + 3)^2 = (4x^2)^2 + 2(4x^2)(3) + 3^2
\]
This simplifies to:
\[
= 16x^4 + 24x^2 + 9
\]
2. **Integrate each term:** Integrate term-by-term:
\[
\int (16x^4 + 24x^2 + 9) \, dx = \int 16x^4 \, dx + \int 24x^2 \, dx + \int 9 \, dx
\]
Calculating each integral:
- \(\int 16x^4 \, dx = \frac{16}{5}x^5\)
- \(\int 24x^2 \, dx = 8x^3\)
- \(\int 9 \, dx = 9x\)
Combine these results:
\[
= \frac{16}{5}x^5 + 8x^3 + 9x + C
\]
where \(C\) is the constant of integration.
3. **Check by differentiation:** Differentiate the result:
\[
\frac{d}{dx}\left(\frac{16}{5}x^5 + 8x^3 + 9x + C\right)
= 16x^4 + 24x^2 + 9
\]
This matches the expanded integrand, confirming our solution is correct.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fee344c66-7efa-4594-9121-d91385396bb3%2Fd60f1722-d006-4f14-a09c-a0226bd0931a%2F5n4f79s_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem:**
Find the indefinite integral and check the results by differentiation.
\[
\int (4x^2 + 3)^2 \, dx
\]
**Solution:**
To solve this problem, we need to follow these steps:
1. **Expand the integrand:** First, expand \((4x^2 + 3)^2\).
Using the formula \((a + b)^2 = a^2 + 2ab + b^2\), we get:
\[
(4x^2 + 3)^2 = (4x^2)^2 + 2(4x^2)(3) + 3^2
\]
This simplifies to:
\[
= 16x^4 + 24x^2 + 9
\]
2. **Integrate each term:** Integrate term-by-term:
\[
\int (16x^4 + 24x^2 + 9) \, dx = \int 16x^4 \, dx + \int 24x^2 \, dx + \int 9 \, dx
\]
Calculating each integral:
- \(\int 16x^4 \, dx = \frac{16}{5}x^5\)
- \(\int 24x^2 \, dx = 8x^3\)
- \(\int 9 \, dx = 9x\)
Combine these results:
\[
= \frac{16}{5}x^5 + 8x^3 + 9x + C
\]
where \(C\) is the constant of integration.
3. **Check by differentiation:** Differentiate the result:
\[
\frac{d}{dx}\left(\frac{16}{5}x^5 + 8x^3 + 9x + C\right)
= 16x^4 + 24x^2 + 9
\]
This matches the expanded integrand, confirming our solution is correct.
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Thank you but I believe you are missing the square in the question
the question is (4x^2 +3)^2 dx
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