Problem 42

Question

a, b, and $c$ are constants and $g(x)$ is a continuous function whose derivative $g^{\prime}(x)$ is also continuous. Use substitution to evaluate the indefinite integrals. $$ \int \frac{g^{\prime}(x)}{[g(x)]^{2}+1} d x $$

Step-by-Step Solution

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Answer

The integral evaluates to $ \tan^{-1}(g(x)) + C $.

1Step 1: Identify the Integral Form

We are given the integral $ \int \frac{g^{\prime}(x)}{[g(x)]^{2}+1} \, dx $. This hints at a substitution method, often used when the derivative of a function is present in the integrand.

2Step 2: Choose a Substitution

Let $ u = g(x) $. Then, the derivative $ du = g^{\prime}(x) \, dx $. This substitution will help us reformulate the integral.

3Step 3: Reformulate the Integral

Substitute $ u = g(x) $ and $ du = g^{\prime}(x) \, dx $ into the integral, transforming it into $ \int \frac{1}{u^2 + 1} \, du $.

4Step 4: Identify the Integral Formula

Recognize $ \int \frac{1}{u^2+1} \, du $ as the standard arctangent, which equals $ \tan^{-1}(u) + C $, where $ C $ is the constant of integration.

5Step 5: Substitute Back to Original Variable

Replace $ u $ with $ g(x) $, our original substitution, yielding $ \tan^{-1}(g(x)) + C $.

Key Concepts

Indefinite IntegralsContinuous FunctionsDerivative Substitution

Indefinite Integrals

Indefinite integrals are a fundamental concept in calculus. They represent the accumulation of quantities where no specific bounds are given. Instead of a definite value, they provide a general form of an antiderivative with a constant of integration. In mathematical terms, the indefinite integral $ \int f(x) \, dx $ of a function $ f(x) $ is a function $ F(x) $ whose derivative is $ f(x) $.

An indefinite integral does not specify any limits of integration.
It includes a constant of integration, usually denoted as $ C $, representing the family of all possible functions that satisfy the derivative $ f(x) $.
Indefinite integrals are often used to reverse differentiation, providing solutions that describe a wide set of potential original functions given their derivatives.

The task for students might involve observing the patterns in the function and its derivative to identify when substitution can simplify the integral calculation. For example, recognizing that $ g'(x) $ is part of the form suggests substitution might be effective. In our exercise, the indefinite integral $ \int \frac{g^{\prime}(x)}{[g(x)]^{2}+1} \, dx $ rallied around this concept to simplify the integration process.

Continuous Functions

Continuous functions are crucial in calculus because they ensure smooth curves without gaps or abrupt changes. This quality allows the use of various calculus operations, such as differentiation and integration, without unexpected complications.

A function $ g(x) $ is continuous if, intuitively, you can draw it without lifting your pencil off the paper.
Mathematically, $ g(x) $ is continuous at point $ a $ if the limit of $ g(x) $ as $ x $ approaches $ a $ equals $ g(a) $.
Continuous derivatives, like $ g'(x) $, enable reliable prediction of changes and applicability of calculus rules like the Fundamental Theorem of Calculus.

In our exercise, $ g(x) $ being continuous and having a continuous derivative $ g^{\prime}(x) $ assures that substitution and integration can proceed smoothly. This guarantees that the manipulation of integrals is valid across all $ x $-values involved, leading to accurate results like $ \tan^{-1}(g(x)) + C $. Such functions behave predictably when subjected to calculus operations, contributing to the predictably smooth transition from $ g(x) $ to $ g'(x) $ and back through integration.

Derivative Substitution

Derivative substitution is an essential technique for simplifying the process of finding indefinite integrals, especially when functions and their derivatives appear together in the integrand. Substitution transforms a complex integral into a simpler one, often by changing variables.

Identify a part of the integrand that is the derivative of another function present. This step is crucial to facilitate substitution.
Choose a substitution, such as $ u = g(x) $, to simplify the integral using the differential $ du = g^{\prime}(x) \, dx $.
Perform the substitution to reformulate the integral, transforming it into a more manageable form, such as $ \int \frac{1}{u^2 + 1} \, du $.
Recognize the simplified integral form for integration. In this example, $ \int \frac{1}{u^2+1} \, du $ is known, corresponding to $ \tan^{-1}(u) + C $.
Finally, replace the substitution variable back to the original variable to complete the integration as $ \tan^{-1}(g(x)) + C $.

This substitution method efficiently handles complex integrands by utilizing known integral forms and leveraging the relationships between functions and their derivatives. It encourages recognizing patterns and inventively reformulating problems, a powerful skill in calculus for smoothing out seemingly complicated integrations.

Problem 42

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