Chapter 6: Problem 40
In Exercises \(25-46,\) use substitution to evaluate the integral. $$\int \frac{x d x}{x^{2}+1}$$
Short Answer
Expert verified
The integral of \(\int \frac{x dx}{x^{2}+1}\) is \(0.5*ln|x^{2}+1|\).
Step by step solution
01
Identify the substituion
Begin by identifying the substitution. Here, it is logical to pick the denominator \(x^{2} + 1\) as the \(u\). Thus, \(u = x^{2} + 1\).
02
Calculate the derivative of u
Differentiate \(u\) with respect to \(x\) to get \(du/dx\). The derivative of \(u = x^{2} + 1\) with respect to \(x\) is \(2x\).
03
Express dx in terms of du
From the second step, we have \(du = 2x dx\). Solving for \(dx\), we get \(dx=du/(2x)\).
04
Substitute in the integral
Substituting \(u\) and \(du\) into the integral, \(\int \frac{x dx}{x^{2}+1}\), yields \(\int \frac{x du}{2xu}\). Note that the \(x\) terms cancel.
05
Simplify and integrate
After cancelling, the integrand is now \(\int \frac{1}{2u} du\). Now it is a simple integral to compute. Integrating \(1/(2u)\) with respect to u gives \(0.5*ln |u|\).
06
Return to the original variable
To finish, substitute the original expression for \(u\): \(u = x^{2} + 1\). Thus, the evaluated integral is \(0.5*ln|x^{2}+1|\).
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with Vaia!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Integration by Substitution
Integration by substitution is a technique used to simplify complex integrals. It involves replacing a part of the integral with a new variable, which makes the integral easier to evaluate. In the provided exercise, substitution is applied by letting a new variable, typically denoted by u, represent a portion of the integrand, which is the function we are integrating.
In this context, the choice of substitution is often guided by the presence of a function and its derivative within the integral. For the exercise \(\int \frac{x dx}{x^{2}+1}\), the denominator \(x^{2} + 1\) suggests a natural choice for our substitution where \(u = x^{2} + 1\). Once the substitution is made, the integral is re-expressed in terms of this new variable u, which often leads to a simpler form that can be integrated more easily.
In this context, the choice of substitution is often guided by the presence of a function and its derivative within the integral. For the exercise \(\int \frac{x dx}{x^{2}+1}\), the denominator \(x^{2} + 1\) suggests a natural choice for our substitution where \(u = x^{2} + 1\). Once the substitution is made, the integral is re-expressed in terms of this new variable u, which often leads to a simpler form that can be integrated more easily.
Indefinite Integral
An indefinite integral represents the family of all antiderivatives of a given function. When we integrate a function without specific limits, the result is an indefinite integral, which includes a constant of integration to account for all potential antiderivatives.
The integral in our exercise, \(\int \frac{x dx}{x^{2}+1}\), is indefinite, as it lacks specific bounds or limits. After performing integration, we arrive at a general solution that captures all possible antiderivatives. In our specific problem, the indefinite integral, after using substitution and simplifying, is expressed as \(0.5*ln|x^{2}+1| + C\), where C represents the constant of integration.
The integral in our exercise, \(\int \frac{x dx}{x^{2}+1}\), is indefinite, as it lacks specific bounds or limits. After performing integration, we arrive at a general solution that captures all possible antiderivatives. In our specific problem, the indefinite integral, after using substitution and simplifying, is expressed as \(0.5*ln|x^{2}+1| + C\), where C represents the constant of integration.
Derivative of a Function
The derivative of a function measures the rate at which the function's output value changes as its input value changes. Mathematically, it represents the slope of the tangent line to the function's graph at any given point.
During the substitution process in integration, we calculate the derivative of the substitution variable, u, with respect to x to accommodate for the different rates of change. In the given example, we determine the derivative of \(u = x^{2} + 1\) to be \(du/dx = 2x\). This step is crucial as it allows us to translate \(dx\) into \(du\) terms, maintaining the equivalent expression under the integral while effectively facilitating substitution.
During the substitution process in integration, we calculate the derivative of the substitution variable, u, with respect to x to accommodate for the different rates of change. In the given example, we determine the derivative of \(u = x^{2} + 1\) to be \(du/dx = 2x\). This step is crucial as it allows us to translate \(dx\) into \(du\) terms, maintaining the equivalent expression under the integral while effectively facilitating substitution.
Simplifying Integrals
Simplifying integrals can be essential for solving them effectively. The process often includes algebraic manipulation, trigonometric identities, or other mathematical properties that make the integrand easier to integrate.
In our example, after substitution and obtaining \(\int \frac{x du}{2xu}\), we notice the opportunity to cancel terms out. Simplifying by canceling x in the numerator and the denominator, we are left with \(\int \frac{1}{2u} du\), which is much simpler to integrate. Such simplifications are a vital step since they can transform a complex integral into a form that is straightforward to evaluate, ultimately leading us to find the indefinite integral more efficiently.
In our example, after substitution and obtaining \(\int \frac{x du}{2xu}\), we notice the opportunity to cancel terms out. Simplifying by canceling x in the numerator and the denominator, we are left with \(\int \frac{1}{2u} du\), which is much simpler to integrate. Such simplifications are a vital step since they can transform a complex integral into a form that is straightforward to evaluate, ultimately leading us to find the indefinite integral more efficiently.
