Problem 408 - Calculus Volume 1

Question

In the following exercises, solve for the antiderivative \(\int f\) of \(f\) with \(C=0,\) then use a calculator to graph \(f\) and the antiderivative over the given interval \([a, b] .\) Identify a value of \(C\) such that adding \(C\) to the antiderivative recovers the definite integral \(F(x)=\int_{a}^{x} f(t) d t .\) $$[\mathbf{T}] \int \frac{9}{9+x^{2}} d x \text { over } [-6,6]$$

Step-by-Step Solution

Answer

Antiderivative: \(3 \tan^{-1} \left(\frac{x}{3}\right)\), \(C=0\).

Step 1: Recognize the form of the integral

The integral \( \int \frac{9}{9+x^2} \, dx \) can be rewritten as \( 9 \cdot \int \frac{1}{9+x^2} \, dx \). This resembles the standard integral form \( \int \frac{1}{a^2+x^2} \, dx = \frac{1}{a} \tan^{-1} \left( \frac{x}{a} \right) + C \). Here, \( a = 3 \).

Step 2: Solve the indefinite integral

Using the atandard integral formula, we solve \( \int \frac{9}{9+x^2} \, dx = 9 \cdot \frac{1}{3} \tan^{-1} \left( \frac{x}{3} \right) + C = 3 \tan^{-1} \left( \frac{x}{3} \right) + C \).

Step 3: Set integration constant C=0

Since the problem specifies to compute the antiderivative with \( C = 0 \), our antiderivative becomes \( 3 \tan^{-1} \left( \frac{x}{3} \right) \).

Step 4: Compute the definite integral for [a,b]

We need to compute the definite integral \( F(x) = \int_{-6}^{6} \frac{9}{9+t^2} \, dt \). This is \( [3 \tan^{-1} ( \frac{x}{3} )]_{-6}^{6} \).

Step 5: Evaluate definite integral

Evaluate at the limits: \( F(x) = 3 \tan^{-1} \left( \frac{6}{3} \right) - 3 \tan^{-1} \left( \frac{-6}{3} \right) = 3 \tan^{-1}(2) - 3 \tan^{-1}(-2) \). The tangent inverse function is odd, meaning \( \tan^{-1}(-x) = -\tan^{-1}(x) \), so this becomes \( 6 \tan^{-1}(2) \).

Step 6: Identify the constant C

The result of the definite integral should match what we would calculate by setting the upper bound to \( x \) and comparing results. Hence, choose \( C \) such that the total integral calculates (use calculator) \( \approx 6 \tan^{-1}(2) \). Let \( C = 0 \) suffice here since it reflects the integral evaluation properly for \( F(x) \).

Key Concepts

Definite Integral

A definite integral is a fundamental concept in calculus used to determine the total accumulation of a quantity. It allows us to find the total amount, area, or change between two specific points. When calculating a definite integral, we integrate a function over an interval \([a, b]\), where:

• \(a\) and \(b\) are the bounds or limits of integration.
• The calculation involves evaluating the antiderivative at these bounds and finding the difference.

This results in a numerical value that represents the area under a curve or other total change.
In our context, we evaluated the definite integral of \( \int_{-6}^6 \frac{9}{9+x^2} \, dx \) to find the total effect of the function from \(-6\) to \(6\). Understanding definite integrals helps solve real-world problems, like finding distance traveled over a certain time or computing the total growth of an investment over time.

Integration Constant

The integration constant, often denoted as \(C\), plays an important role in calculating indefinite integrals. When calculating an indefinite integral, we find the antiderivative and include \(C\) to represent any possible shifts in the function:

• Every function has an infinite number of antiderivatives differing by a constant \(C\).
• The integration constant ensures the entire family of possible solutions is covered.

For definite integrals, the \(C\) cancels out, as it involves a specific interval, but it’s crucial in solving indefinite integrals. In our exercise, setting \( C = 0 \) simplifies the calculation, allowing immediate focus on the principal value of the antiderivative without any shift.

Inverse Tangent Function

The inverse tangent function, written as \( \tan^{-1}(x) \) or \ \text{arctan}(x) \, is one of the six inverse trigonometric functions, essential in many integration processes. It helps relate angles to their corresponding tangent values:

• The inverse tangent function returns an angle whose tangent is \(x\).
• Its range is typically between \(-\frac{\pi}{2}\) and \(\frac{\pi}{2}\).
• It is an odd function, meaning \( \tan^{-1}(-x) = -\tan^{-1}(x) \).

In our problem, applying the inverse tangent function is crucial to solving the integral \(( \int \frac{9}{9+x^2} \, dx )\), using the known identity for easier evaluation.

Standard Integral Forms

Standard integral forms are pre-established formulas for integrating common functions, saving time and reducing errors when solving integrals. These forms include solutions for basic trigonometric, polynomial, and rational functions:

• They act as a reference, letting you quickly match and solve integrals.
• Using them correctly requires understanding the form and similarity of the integral you have.

In our example, we used the standard integral form \( \int \frac{1}{a^2+x^2} \, dx = \frac{1}{a} \tan^{-1} \left( \frac{x}{a} \right) + C \) to solve the integral easily. Recognizing and applying these forms is a vital skill for any calculus student, simplifying many integration tasks by avoiding lengthy integration by parts or substitution methods.