Problem 304 - Calculus Volume 1

Question

In the following exercises, evaluate the indefinite integral \(\int f(x) d x\) with constant \(C=0\) using u-substitution. Then, graph the function and the antiderivative over the indicated interval. If possible, estimate a value of \(C\) that would need to be added to the antiderivative to make it equal to the definite integral \(F(x)=\int_{a}^{x} f(t) d t,\) with a the left endpoint of the given interval. If \(h(a)=h(b)\) in \(\int_{a}^{b} g^{\prime} h(x) h_{h}(x) d x,\) what can you say about the value of the integral?

Step-by-Step Solution

Answer

Use u-substitution to find the antiderivative, compare for \(C\), and the integral may evaluate to zero if boundary values of \(h\) are equal.

Step 1: Understand the Problem

We need to evaluate the indefinite integral of a given function using u-substitution with constant \(C=0\). After that, graph the function and its antiderivative over a specified interval. Lastly, estimate a constant \(C\) to make the antiderivative equal to a definite integral \(F(x)\). We then consider what happens to a definite integral with \(h(a) = h(b)\).

Step 2: Set up the Integral for U-Substitution

Identify the function \(f(x)\) from the integral \(\int f(x) \, dx\). Choose a substitution \(u = g(x)\) such that the derivative \(\frac{du}{dx} = g'(x)\) can replace part of \(f(x)\) in terms of \(u\). Integrate with respect to \(u\).

Step 3: Integrate and Substitute Back

Perform the integration to obtain the antiderivative with respect to \(u\), and then substitute back the expression of \(x\) to get the antiderivative in terms of \(x\).

Step 4: Graph the Functions

Graph both the original function \(f(x)\) and its antiderivative over the given interval. Visually estimate where a possible constant might adjust the indefinite integral to match a definite integral over this interval.

Step 5: Estimate the Constant C

Calculate the definite integral \(F(x) = \int_{a}^{x} f(t) \, dt\) and compare it to the antiderivative \(F(x) + C\). Adjust and estimate the value of \(C\) such that the antiderivative equals the definite integral over the interval.

Step 6: Consider the Special Condition

Given \(h(a) = h(b)\) for the integral \(\int_{a}^{b} g'(x) h(x) h'(x) \, dx\), the expression suggests that the terms of the integrand might cancel out over the interval, potentially making the integral evaluate to zero.

Key Concepts

Indefinite Integral

An indefinite integral, also known as an antiderivative, is a fundamental concept in calculus. It represents the inverse operation of differentiation. When we integrate a function without specific limits, we obtain a family of functions described as the antiderivatives of the original function. This process involves finding a function whose derivative is the given function. For example, if we have a function \( f(x) \), its indefinite integral is denoted by

• \( \int f(x) \, dx \)

Finding the indefinite integral involves adding a constant \( C \) because the process of differentiation deletes any constant term. Hence, the indefinite integral represents a whole family of functions shifting vertically by different values of \( C \).

Definite Integral

A definite integral differs from an indefinite integral by including limits of integration, which specifies a particular interval over which the function is integrated. It calculates the net area under the curve of the function between two specified points. A definite integral is represented as:

• \( \int_{a}^{b} f(x) \, dx \)

This not only provides a number, known as the integral’s value, but also gives a physical meaning, such as total accumulation or the area between the curve and the x-axis. In the context of comparing with an indefinite integral, choosing an appropriate \( C \) can help make a particular antiderivative provide this same numeric result across the specified interval [a, b].

Antiderivative

An antiderivative of a function \( f(x) \) is a function \( F(x) \) such that the derivative of \( F(x) \) is equal to \( f(x) \). Formally, if \( f'(x) = F(x) \), then \( F(x) \) is called an antiderivative of \( f(x) \). When computing an antiderivative, a constant \( C \) is added because differentiation erases constants. Thus,

• \( F(x) + C = \int f(x) \, dx \)

obtains an entire family of functions for \( F(x) \). In exercises, once you identify the antiderivative, you can further adjust with \( C \) to fit particular conditions, such as matching definite integrals over an interval.

Integration Techniques

Integration techniques are various methods used to determine the integral of functions. One of the primary techniques is u-substitution, which simplifies the function into a new variable, \( u \), by making a suitable choice \( u = g(x) \). Then, calculate \( \frac{du}{dx} \), which lets you express \( dx \) in terms of \( du \).

After replacing \( f(x) \) with expressions in terms of \( u \), you perform the integration. Once integrated, it is crucial to substitute back to the original variable, \( x \), to get the final antiderivative in terms of \( x \).

• U-substitution is particularly useful when the integral is composed of a function and its derivative.
• It is similar to the reverse process of the chain rule in differentiation.

These techniques expand your toolbox for solving integrals, making complex problems more approachable by converting them into simpler forms.