10 Consider the function f(x) = 5x¹0 + 7x³ – 6x¹ – 4. Enter an antiderivative of f(x). Do not enter +c as part of your answer. Answer: +c
10 Consider the function f(x) = 5x¹0 + 7x³ – 6x¹ – 4. Enter an antiderivative of f(x). Do not enter +c as part of your answer. Answer: +c
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![**Problem Statement:**
Consider the function \( f(x) = 5x^{10} + 7x^5 - 6x^4 - 4 \).
Enter an antiderivative of \( f(x) \). Do not enter \( +c \) as part of your answer.
**Answer:** [ ] \( + c \)
---
**Explanation:**
To find the antiderivative (or indefinite integral) of the function \( f(x) = 5x^{10} + 7x^5 - 6x^4 - 4 \), apply the power rule for integration, which states:
\[
\int x^n \, dx = \frac{x^{n+1}}{n+1} + C
\]
Apply this rule to each term in the function separately:
1. For \( 5x^{10} \):
\[
\int 5x^{10} \, dx = \frac{5x^{11}}{11}
\]
2. For \( 7x^5 \):
\[
\int 7x^5 \, dx = \frac{7x^6}{6}
\]
3. For \( -6x^4 \):
\[
\int -6x^4 \, dx = \frac{-6x^5}{5}
\]
4. For \( -4 \):
\[
\int -4 \, dx = -4x
\]
Combine the results:
\[
F(x) = \frac{5x^{11}}{11} + \frac{7x^6}{6} - \frac{6x^5}{5} - 4x + C
\]
Omit \( +C \) as instructed.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9978ffa8-8e6a-4550-8363-e044b6a6e895%2Fd9b4a863-f5ea-48b7-8b6f-f8d59266b719%2Fvbro5n_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Consider the function \( f(x) = 5x^{10} + 7x^5 - 6x^4 - 4 \).
Enter an antiderivative of \( f(x) \). Do not enter \( +c \) as part of your answer.
**Answer:** [ ] \( + c \)
---
**Explanation:**
To find the antiderivative (or indefinite integral) of the function \( f(x) = 5x^{10} + 7x^5 - 6x^4 - 4 \), apply the power rule for integration, which states:
\[
\int x^n \, dx = \frac{x^{n+1}}{n+1} + C
\]
Apply this rule to each term in the function separately:
1. For \( 5x^{10} \):
\[
\int 5x^{10} \, dx = \frac{5x^{11}}{11}
\]
2. For \( 7x^5 \):
\[
\int 7x^5 \, dx = \frac{7x^6}{6}
\]
3. For \( -6x^4 \):
\[
\int -6x^4 \, dx = \frac{-6x^5}{5}
\]
4. For \( -4 \):
\[
\int -4 \, dx = -4x
\]
Combine the results:
\[
F(x) = \frac{5x^{11}}{11} + \frac{7x^6}{6} - \frac{6x^5}{5} - 4x + C
\]
Omit \( +C \) as instructed.
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