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Magazine Chapter 7: Problem 55
Find the general antiderivative. Check your answers by differentiation. $$f(x)=\frac{x}{x^{2}+1}$$
Short Answer
Expert verified
The general antiderivative is \( \frac{1}{2} \ln(x^2 + 1) + C \).
Step by step solution
01
Identify the function needing integration
We have the function \( f(x) = \frac{x}{x^2 + 1} \). Our task is to find the general antiderivative of this function.
02
Recognize substitution method
Notice that the denominator of the integrand, \( x^2 + 1 \), has a simple derivative, which is \( 2x \). This suggests that a substitution method might work well. We can let \( u = x^2 + 1 \), which simplifies the problem.
03
Perform substitution
Set \( u = x^2 + 1 \). Then, the differential becomes \( du = 2x \, dx \), which implies \( \frac{du}{2} = x \, dx \). The integral now becomes \[ \int \frac{x}{x^2+1} \, dx = \int \frac{1}{u} \cdot \frac{du}{2} \].
04
Integrate with respect to \( u \)
The integral in terms of \( u \) is \( \int \frac{1}{u} \cdot \frac{du}{2} = \frac{1}{2} \int \frac{1}{u} \, du \). This integrates to \( \frac{1}{2} \ln |u| + C \), where \( C \) is the constant of integration.
05
Substitute back to \( x \)
Return to the variable \( x \) by substituting \( u \) back: \( u = x^2 + 1 \). The antiderivative is \( \frac{1}{2} \ln |x^2 + 1| + C \). Under real number conditions, since \( x^2 + 1 > 0 \) for all \( x \), the absolute value can be dropped.
06
Differentiate to check your solution
Differentiate the antiderivative, \( \frac{1}{2} \ln (x^2 + 1) + C \). Using the chain rule: \[ \frac{d}{dx} \left( \frac{1}{2} \ln(x^2 + 1) \right) = \frac{1}{2} \cdot \frac{1}{x^2 + 1} \cdot 2x = \frac{x}{x^2 + 1} \]. This matches \( f(x) \), confirming the antiderivative is correct.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Substitution Method
The substitution method is a powerful technique in integration, allowing us to simplify complex integrals. It's especially handy when you recognize that part of the integrand resembles the derivative of another function.
In this exercise, the substitution method is applied by letting \( u = x^2 + 1 \). This matches the denominator's derivative closely, which is \( 2x \). By taking this approach:- Substitute \( u = x^2 + 1 \), transforming the integral.- Differentiate to find \( du = 2x \, dx \), re-arranged as \( \frac{du}{2} = x \, dx \).- Simplify the integral to easier terms in \( u \).This method dramatically reduces complexity, making it pivotal for finding antiderivatives more efficiently.
In this exercise, the substitution method is applied by letting \( u = x^2 + 1 \). This matches the denominator's derivative closely, which is \( 2x \). By taking this approach:- Substitute \( u = x^2 + 1 \), transforming the integral.- Differentiate to find \( du = 2x \, dx \), re-arranged as \( \frac{du}{2} = x \, dx \).- Simplify the integral to easier terms in \( u \).This method dramatically reduces complexity, making it pivotal for finding antiderivatives more efficiently.
Integration
Integration is the process of finding antiderivatives, the opposite operation of differentiation. When we integrate, we effectively "undo" the derivative, searching for the original function.
In our function \( f(x) = \frac{x}{x^2 + 1} \), the integration process involves simplifying the integrand through substitution. This allows us to perform the integration:- The integral becomes \( \frac{1}{2} \int \frac{1}{u} \, du \).- This is recognized as a standard natural logarithm integral.The integration results in \( \frac{1}{2} \ln |u| + C \), where \( C \) is the constant of integration, representing an infinite family of solutions.
In our function \( f(x) = \frac{x}{x^2 + 1} \), the integration process involves simplifying the integrand through substitution. This allows us to perform the integration:- The integral becomes \( \frac{1}{2} \int \frac{1}{u} \, du \).- This is recognized as a standard natural logarithm integral.The integration results in \( \frac{1}{2} \ln |u| + C \), where \( C \) is the constant of integration, representing an infinite family of solutions.
Differentiation
Differentiation is the process of calculating a derivative, which represents the rate of change of a function. It's the reverse operation of integration. After finding the antiderivative of a function, checking it via differentiation confirms its correctness.
Here, the differentiation step serves this purpose:- Differentiate \( \frac{1}{2} \ln(x^2 + 1) + C \).- Use the chain rule to find the derivative.The result, \( \frac{x}{x^2 + 1} \), matches the original function \( f(x) \), verifying our antiderivative.
Here, the differentiation step serves this purpose:- Differentiate \( \frac{1}{2} \ln(x^2 + 1) + C \).- Use the chain rule to find the derivative.The result, \( \frac{x}{x^2 + 1} \), matches the original function \( f(x) \), verifying our antiderivative.
Chain Rule
The chain rule is essential in both differentiation and integration, helping us manage composite functions. It's particularly useful when differentiating complex expressions.
In this context, the chain rule aids in differentiating the natural logarithm function we obtained:- Inside our function, we have \( \, \ln(x^2 + 1) \).- The chain rule states: \( \frac{d}{dx} [\ln(u)] = \frac{1}{u} \cdot \frac{du}{dx} \).Applying the chain rule results in \( \frac{1}{2} \cdot \frac{1}{x^2+1} \cdot 2x = \frac{x}{x^2 + 1} \), confirming our solution through differentiation.
In this context, the chain rule aids in differentiating the natural logarithm function we obtained:- Inside our function, we have \( \, \ln(x^2 + 1) \).- The chain rule states: \( \frac{d}{dx} [\ln(u)] = \frac{1}{u} \cdot \frac{du}{dx} \).Applying the chain rule results in \( \frac{1}{2} \cdot \frac{1}{x^2+1} \cdot 2x = \frac{x}{x^2 + 1} \), confirming our solution through differentiation.
