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Chapter 4: Problem 22

\(21-22\) Find the antiderivative \(F\) of \(f\) that satisfies the given condition. Check your answer by comparing the graphs of \(f\) and \(F .\) \(f(x)=4-3\left(1+x^{2}\right)^{-1}, \quad F(1)=0\)

Short Answer

Expert verified
The antiderivative is \( F(x) = 4x - 3\text{atan}(x) + \frac{3\pi}{4} - 4 \).

Step by step solution

01

Understand the Problem

The function given is \( f(x) = 4 - 3(1 + x^2)^{-1} \) and we need to find its antiderivative \( F(x) \) such that \( F(1) = 0 \). This means we are looking for a function whose derivative is \( f(x) \) and meets the condition \( F(1) = 0 \).
02

Antiderivative of Constant Term

Start by integrating the constant \( 4 \). The antiderivative of a constant \( c \) is \( cx + C \), where \( C \) is a constant of integration. Thus, the antiderivative of \( 4 \) is \( 4x \).
03

Antiderivative of Rational Function

Next, deal with the second part \( -3(1+x^2)^{-1} \). This resembles the derivative of \( \text{atan}(x) \), since \( \frac{d}{dx} [\text{atan}(x)] = \frac{1}{1+x^2} \). Therefore, the antiderivative of \( -3(1+x^2)^{-1} \) is \( -3\text{atan}(x) \).
04

Combine Antiderivatives

Combine the antiderivatives from Steps 2 and 3: \( F(x) = 4x - 3\text{atan}(x) + C \) where \( C \) is a constant.
05

Use Given Condition

Apply the condition \( F(1) = 0 \) to solve for \( C \). Substitute \( x = 1 \) into the equation: \( 0 = 4(1) - 3\text{atan}(1) + C \). \( \text{atan}(1) = \frac{\pi}{4} \), so we have \( 0 = 4 - \frac{3\pi}{4} + C \) Thus, \( C = \frac{3\pi}{4} - 4 \).
06

Final Antiderivative

Substitute \( C \) back into the function to find the final form of \( F(x) \): \( F(x) = 4x - 3\text{atan}(x) + \frac{3\pi}{4} - 4 \).
07

Verify by Graph

Compare the graphs of \( f(x) = 4 - 3(1+x^2)^{-1} \) and \( F(x) = 4x - 3\text{atan}(x) + \frac{3\pi}{4} - 4 \). Ensure that the slope of \( F(x) \) at any point is equal to the value of \( f(x) \), confirming that \( F'(x) = f(x) \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Integral Calculus
Integral calculus is all about finding the antiderivatives, or integrals, of functions. The antiderivative, often denoted as \( F(x) \), is a function whose derivative is the given function \( f(x) \). In this exercise, we are asked to find the antiderivative of the function \( f(x) = 4 - 3(1 + x^2)^{-1} \).
  • The first step is to integrate each part of the function separately. For the constant term \(4\), the antiderivative is straightforward. It is \(4x + C\), where \(C\) is the constant of integration that we will solve later.
  • The more complex part \(-3(1 + x^2)^{-1}\) resembles the derivative of \(\text{atan}(x)\), which means its antiderivative is \(-3 \cdot \text{atan}(x)\).
The challenge in integral calculus exercises like this is combining these parts and using any given conditions to find the entire antiderivative function.
Trigonometric Function
Trigonometric functions can often appear in unexpected places within calculus. In this particular exercise, the function \(\text{atan}(x)\) is involved. The arctangent function, which is commonly seen in integration, comes from recognizing the form \((1 + x^2)^{-1}\).
  • Integration of \((1+x^2)^{-1}\) is related to the derivative of \(\text{atan}(x)\): \(\frac{d}{dx} [\text{atan}(x)] = \frac{1}{1+x^2}\).
  • By multiplying it with \(-3\), the antiderivative becomes \(-3 \cdot \text{atan}(x)\).
Recognizing such forms and their associated trigonometric functions can simplify solving integrals, making direct connections between derivatives and antiderivatives more apparent.
Initial Condition
The initial condition is crucial in determining the constant term \(C\) when finding antiderivatives. Here, we have the condition \(F(1) = 0\), which tells us that when \(x = 1\), the function \(F(x)\) must equal zero. This initial condition helps to find the exact value of \(C\).
To solve for \(C\), substitute \(x = 1\) into the expression for \(F(x)\):
  • Start with \(F(x) = 4x - 3\text{atan}(x) + C\).
  • Since \(\text{atan}(1) = \frac{\pi}{4}\), substitute to find \(0 = 4(1) - \frac{3\pi}{4} + C\).
  • Solve the equation to find \(C = \frac{3\pi}{4} - 4\).
Including the initial condition ensures that the function \(F(x)\) satisfies all the requirements of the problem, providing a specific solution that fits the given condition.

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