1. Find the most general antiderivative or indefinite integral for the function. (Ilint: Using a trigonometric identity might be helpful). I cot² (5) de
1. Find the most general antiderivative or indefinite integral for the function. (Ilint: Using a trigonometric identity might be helpful). I cot² (5) de
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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![1. Find the most general antiderivative or indefinite integral for the function. *(Hint: Using a trigonometric identity might be helpful).*
\[
\int \cot^2(5x) \, dx
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa161a3a5-b659-479f-b9ff-bebdce9b53cf%2F272f26b8-7cc6-471e-becf-a00d29ed3305%2Fmiy4i1s_processed.jpeg&w=3840&q=75)
Transcribed Image Text:1. Find the most general antiderivative or indefinite integral for the function. *(Hint: Using a trigonometric identity might be helpful).*
\[
\int \cot^2(5x) \, dx
\]
![**Problem Statement:**
2. Find a function \( f \) such that \( f'(x) = 4x^2 + 5x - 2 \) **AND** \( f(0) = 4 \).
---
**Solution Approach:**
To find the function \( f(x) \), we start by integrating the derivative \( f'(x) \).
1. **Integrate \( f'(x) \):**
\[
f(x) = \int (4x^2 + 5x - 2) \, dx
\]
- The integral of \( 4x^2 \) is \(\frac{4}{3}x^3\).
- The integral of \( 5x \) is \(\frac{5}{2}x^2\).
- The integral of \(-2\) is \(-2x\).
- Don't forget to add the constant of integration \( C \).
So, the integrated function becomes:
\[
f(x) = \frac{4}{3}x^3 + \frac{5}{2}x^2 - 2x + C
\]
2. **Apply the Initial Condition \( f(0) = 4 \):**
Substitute \( x = 0 \) in the function to find \( C \):
\[
4 = \frac{4}{3}(0)^3 + \frac{5}{2}(0)^2 - 2(0) + C \Rightarrow C = 4
\]
Therefore, the function \( f(x) \) is:
\[
f(x) = \frac{4}{3}x^3 + \frac{5}{2}x^2 - 2x + 4
\]
This answer satisfies both conditions given in the problem statement: \( f'(x) = 4x^2 + 5x - 2 \) and \( f(0) = 4 \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa161a3a5-b659-479f-b9ff-bebdce9b53cf%2F272f26b8-7cc6-471e-becf-a00d29ed3305%2Fbuabvwr_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
2. Find a function \( f \) such that \( f'(x) = 4x^2 + 5x - 2 \) **AND** \( f(0) = 4 \).
---
**Solution Approach:**
To find the function \( f(x) \), we start by integrating the derivative \( f'(x) \).
1. **Integrate \( f'(x) \):**
\[
f(x) = \int (4x^2 + 5x - 2) \, dx
\]
- The integral of \( 4x^2 \) is \(\frac{4}{3}x^3\).
- The integral of \( 5x \) is \(\frac{5}{2}x^2\).
- The integral of \(-2\) is \(-2x\).
- Don't forget to add the constant of integration \( C \).
So, the integrated function becomes:
\[
f(x) = \frac{4}{3}x^3 + \frac{5}{2}x^2 - 2x + C
\]
2. **Apply the Initial Condition \( f(0) = 4 \):**
Substitute \( x = 0 \) in the function to find \( C \):
\[
4 = \frac{4}{3}(0)^3 + \frac{5}{2}(0)^2 - 2(0) + C \Rightarrow C = 4
\]
Therefore, the function \( f(x) \) is:
\[
f(x) = \frac{4}{3}x^3 + \frac{5}{2}x^2 - 2x + 4
\]
This answer satisfies both conditions given in the problem statement: \( f'(x) = 4x^2 + 5x - 2 \) and \( f(0) = 4 \).
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