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Chapter 7: Problem 60

(a) Make an appropriate \(u\) -substitution of the form \(u=x^{1 / n}\) or \(u=(x+a)^{1 / n}\), and then evaluate the integral. (b) If you have a CAS, use it to evaluate the integral, and then confirm that the result is equivalent to the one that you found in part (a). $$ \int \frac{\sqrt{x}}{x+1} d x $$

Short Answer

Expert verified
The integral evaluates to \( 2(\sqrt{x} - \tan^{-1}(\sqrt{x})) + C \).

Step by step solution

01

Identify the Substitution

We are given the integral \( \int \frac{\sqrt{x}}{x+1} \, dx \). We will make a substitution to simplify the integral. A substitution of the form \( u = x^{1/2} \) or \( u = \sqrt{x} \) will be useful here, as it matches the form of the integrand.
02

Perform the Substitution

Set \( u = \sqrt{x} \). Then, \( x = u^2 \) and \( dx = 2u \, du \). Substitute these into the integral to get:\[ \int \frac{\sqrt{x}}{x+1} \, dx = \int \frac{u}{u^2 + 1} \, 2u \, du = 2 \int \frac{u^2}{u^2 + 1} \, du \]
03

Simplify the Integral

The integral becomes \( 2 \int \left(1 - \frac{1}{u^2 + 1}\right) \, du \). We can split it into two separate integrals:\[ 2 \left( \int 1 \, du - \int \frac{1}{u^2 + 1} \, du \right) \]
04

Integrate Each Part

Now, calculate each integral separately:- \( \int 1 \, du = u \)- \( \int \frac{1}{u^2 + 1} \, du = \tan^{-1}(u) \)Substitute back into the main integral:\[ 2 \left( u - \tan^{-1}(u) \right) + C \]
05

Back-Substitute the Variable

Recall that \( u = \sqrt{x} \). Substitute back to get the integral in terms of \( x \):\[ 2 \left( \sqrt{x} - \tan^{-1}(\sqrt{x}) \right) + C \]
06

Confirm with CAS

Use a computer algebra system (CAS) to evaluate the integral \( \int \frac{\sqrt{x}}{x+1} \, dx \). The CAS should confirm that the antiderivative is equivalent to:\[ 2 \left( \sqrt{x} - \tan^{-1}(\sqrt{x}) \right) + C \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding u-substitution
Integral calculus can be challenging, but with the right techniques, it becomes more approachable. One such method is u-substitution. This technique is used to simplify integrals by substituting a part of the integrand with a new variable, usually denoted as \( u \). Let's break it down:
  • Select the substitution: Choose a part of the integrand to become new variable \( u \). For example, in the integral \( \int \frac{\sqrt{x}}{x+1} \, dx \), \( u = \sqrt{x} \).
  • Express \( x \) in terms of \( u \): This involves solving for \( x \) from the chosen \( u \). Here, it becomes \( x = u^2 \).
  • Find \( dx \): Differentiate \( x = u^2 \) to get \( dx = 2u \, du \).
  • Substitute in the integral: Replace \( x \), \( \sqrt{x} \), and \( dx \) in the original integral with the expressions in terms of \( u \). This changes the integral to \( 2 \int \frac{u^2}{u^2 + 1} \, du \).
Using u-substitution effectively transforms the integral into a simpler form, allowing for easier computation and deeper understanding of the function's behavior.
The Role and Calculation of Antiderivatives
After simplifying an integral using techniques like u-substitution, the next step is to find its antiderivative. An antiderivative of a function is essentially its integral, or the function whose derivative leads back to the original function.To compute the antiderivative within our example, you can:
  • Simplify the integral expression: The given example involves the substitution-integrand \( 2 \int \left(1 - \frac{1}{u^2 + 1}\right) \, du \). Split the integral into simpler parts: \( 2 \int 1 \, du - 2 \int \frac{1}{u^2 + 1} \, du \).
  • Integrate each part:
    • Constant terms: \( \int 1 \, du = u \)
    • Arctangent function: \( \int \frac{1}{u^2 + 1} \, du = \tan^{-1}(u) \)
  • Combine the results: Bring these results together to form the complete antiderivative: \( 2 ( u - \tan^{-1}(u) ) + C \).
Antiderivatives provide a valuable insight into the accumulated area under a curve or function, forming a fundamental component of integral calculus.
Exploring Computer Algebra Systems in Integral Calculus
A Computer Algebra System (CAS) is a powerful tool in solving complex calculus problems that would otherwise be tedious and time-consuming. These systems are programmed to perform symbolic mathematics, allowing for the computation of derivatives, integrals, and other algebraic manipulations.In the context of solving integrals:
  • Verification: After manually calculating the antiderivative using techniques like u-substitution, the result can be verified using a CAS. In our example, it confirmed that the antiderivative \( 2 \left( \sqrt{x} - \tan^{-1}(\sqrt{x}) \right) + C \) was correct.
  • Speed and accuracy: CAS offers a quick and accurate solution, reducing human error that could occur in manual calculations.
  • Complex problems: Able to handle more complex integrals that may be challenging to solve by hand, saving time in educational and professional settings.
Overall, a CAS supports learning by providing instant validation and insights into calculus problems, making it an invaluable resource in both academic and practical applications.

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Most popular questions from this chapter

Use any method to solve for \(x\). $$ \int_{1}^{x} \frac{1}{t \sqrt{2 t-1}} d t=1, x>\frac{1}{2} $$

(a) Make an appropriate \(u\) -substitution of the form \(u=x^{1 / n}\) or \(u=(x+a)^{1 / n}\), and then evaluate the integral. (b) If you have a CAS, use it to evaluate the integral, and then confirm that the result is equivalent to the one that you found in part (a). $$ \int \frac{1}{x \sqrt{x^{3}-1}} d x $$

Medication can be administered to a patient using a variety of methods. For a given method, let \(c(t)\) denote the concentration of medication in the patient's bloodstream (measured in \(\mathrm{mg} / \mathrm{L}) t\) hours after the dose is given. The area under the curve \(c=c(t)\) over the time interval \([0,+\infty)\) indicates the "availability" of the medication for the patient's body. Determine which method provides the greater availability. Method \(1: c_{1}(t)=6\left(e^{-0.4 t}-e^{-1.3 t}\right)\) Method \(2: c_{2}(t)=5\left(e^{-0.4 t}-e^{-3 t}\right)\)

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Transform the given improper integral into a proper integral by making the stated \(u\) -substitution; then approximate the proper integral by Simpson's rule with \(n=10\) subdivisions. Round your answer to three decimal places. $$ \int_{0}^{1} \frac{\sin x}{\sqrt{1-x}} d x ; u=\sqrt{1-x} $$
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