Problem 409 - 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 \(F(x)=\int_{a}^{x} f(t) d t\). [T] \(\int \frac{\cos x}{4+\sin ^{2} x} d x\) over [-6,6]

Step-by-Step Solution

Answer

The value of \(C\) maintains the indefinite antiderivative consistent with the definite integral within the given interval.

Step 1: Identify the Integral Problem

We need to find the antiderivative of the expression \(\int \frac{\cos x}{4+\sin^2 x} \, dx\) with the constant of integration \(C = 0\).

Step 2: Use Substitution

To solve \(\int \frac{\cos x}{4+\sin^2 x} \, dx\), use the substitution \(u = \sin x\), which implies \(du = \cos x \, dx\). Thus the integral becomes \(\int \frac{1}{4+u^2} \, du\).

Step 3: Integrate Resulting Expression

The integral \(\int \frac{1}{4+u^2} \, du\) can be recognized as a standard form, resulting in \(\frac{1}{2} \tan^{-1}\left(\frac{u}{2}\right) + C\). Substituting back, we find the antiderivative is \(F(x) = \frac{1}{2} \tan^{-1}\left(\frac{\sin x}{2}\right)\).

Step 4: Graph the Function and Antiderivative

Use a graphing calculator to plot both the function \(f(x) = \frac{\cos x}{4+\sin^2 x}\) and its antiderivative \(F(x) = \frac{1}{2} \tan^{-1}\left(\frac{\sin x}{2}\right)\) over the interval \([-6, 6]\). Ensure that the constant \(C\) is set to zero for the antiderivative plot.

Step 5: Determine Value of C

Evaluate the definite integral \(\int_{-6}^{x} \frac{\cos t}{4+\sin^2 t} \, dt\). Compare this with the graph of the antiderivative already established. Adjust \(C\) such that the antiderivative matches this definite integral. This requires adding \(C\) such that \(F(x) = \int_{-6}^{x} f(t) \, dt\), which in this context is primarily verifying consistency over the interval.

Key Concepts

Antiderivative

An antiderivative of a function is essentially a "reverse" process of differentiation. It is a function whose derivative gives back the original function. When finding the antiderivative, we use an integral sign: \( \int \). This process includes adding a constant \( C \), which accounts for any vertical shift in the graph.
The antiderivative can take numerous forms depending on that constant, which makes it unique to indefinite integrals. In our problem, the task was to find the antiderivative of \( \int \frac{\cos x}{4+\sin^2 x} \, dx \) with \( C=0 \).
Through substitution, this integral simplifies, revealing the antiderivative as \( F(x) = \frac{1}{2} \tan^{-1}\left(\frac{\sin x}{2}\right) \). The goal is not only to determine this function but to understand how it behaves over a range, specifically the interval \([-6, 6]\). By mastering the concept of an antiderivative, students can solve more complex integrals with ease.

Definite Integral

A definite integral is an integral that calculates an area or accumulated quantity within a specific interval \([a, b]\). Unlike an antiderivative, a definite integral evaluates to a specific number rather than a function.
In our exercise, after determining the antiderivative, you use the interval \([-6, 6]\) to evaluate the definite integral \( \int_{-6}^{x} \frac{\cos t}{4+\sin^2 t} \, dt \). The role of the constant \( C \) becomes crucial here, ensuring that the antiderivative aligns with the definite integral across this specific range.
Understanding definite integrals is similar to understanding the net change over an interval, which is foundational for many applications, from physics to statistics.

Trigonometric Substitution

Trigonometric substitution is a technique employed to solve integrals involving expressions like \( \sin x \) or \( \cos x \). It simplifies the integral by using trigonometric identities to transform complex expressions into simpler ones.
In the original problem, the substitution \( u = \sin x \) is used. This substitution allows \( du = \cos x \, dx \), changing the integral into a simpler form: \( \int \frac{1}{4+u^2} \, du \). This is a recognizable standard form integrals, making it much simpler to find the antiderivative.
Trigonometric substitution is a powerful tool, converting difficult integrals into forms that can be evaluated with standard techniques, helping solve problems involving trigonometric functions effectively.

Graphical Analysis

Graphical analysis involves visually interpreting the behavior of functions. For integrals, this can mean examining both a function \( f(x) \) and its antiderivative \( F(x) \) on a graph. In our exercise, you were asked to plot \( f(x) = \frac{\cos x}{4+\sin^2 x} \) and its antiderivative on the interval \([-6, 6]\).
This reveals how the antiderivative grows as it accumulates the area under the curve of the function. By setting \( C = 0 \), you can see the primitive form of the function. Adjusting \( C \) aligns the antiderivative graph with the evaluated exact area from the definite integral.
Graphical analysis provides a deeper understanding by illustrating changes and relationships visually, offering insight that purely algebraic solutions can't always convey.