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Chapter 9: Problem 27

(a) Make the indicated \(u\) -substitution, and then use the Endpaper Integral Table to 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). $$\int \frac{1}{\sqrt{x}(9 x+4)} d x, u=3 \sqrt{x}$$

Short Answer

Expert verified
The integral evaluates to \(3 \tan^{-1}\left(\frac{3\sqrt{x}}{2}\right) + C\).

Step by step solution

01

Perform the u-Substitution

Given the substitution \(u = 3\sqrt{x}\), let's express \(x\) in terms of \(u\). We have \(u = 3\sqrt{x} \Rightarrow \sqrt{x} = \frac{u}{3} \Rightarrow x = \left(\frac{u}{3}\right)^2 = \frac{u^2}{9}\). Differentiate this to find \(dx\): \(dx = \frac{2u}{9}du\).
02

Substitute into the Integral

Substitute \(x\) and \(dx\) in terms of \(u\) into the integral. The original integral becomes:\[ \int \frac{1}{\sqrt{x}(9x + 4)}\, dx = \int \frac{1}{\frac{u}{3}\left(9\left(\frac{u^2}{9}\right) + 4\right)}\cdot \frac{2u}{9}\, du \]Simplifying, we get:\[ \int \frac{6u}{(u)(u^2 + 4)}\, du = \int \frac{6}{u^2 + 4}\, du \]
03

Evaluate the Integral

The integral \( \int \frac{6}{u^2 + 4} \ du \) can be solved using the formula for integrals of the form \( \int \frac{1}{a^2 + b^2} \ db = \frac{1}{a} \tan^{-1}\left(\frac{b}{a}\right) + C \). Here, \(a = 2\). Thus, the integral is:\[ 3 \tan^{-1}\left(\frac{u}{2}\right) + C \]
04

Back-Substitute for x

Recall that \(u = 3\sqrt{x}\). Substitute back to express the solution in terms of \(x\):\[ 3 \tan^{-1}\left(\frac{3\sqrt{x}}{2}\right) + C \]
05

Use a CAS to Confirm the Solution

Use a computer algebra system (CAS) to evaluate the integral \( \int \frac{1}{\sqrt{x}(9x + 4)}\, dx \). The CAS should provide the same result, confirming that the solution \(3 \tan^{-1}\left(\frac{3\sqrt{x}}{2}\right) + C\) is indeed correct, verifying the calculation done by hand.

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Key Concepts

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

Understanding u-Substitution
U-substitution is a powerful technique in integration, especially useful when dealing with complex expressions. We use it to transform a challenging integral into a simpler one by taking advantage of a substitution variable, often referred to as "u." This process involves:

  • Identifying a part of the integrand to substitute with "u."
  • Expressing other terms of the integrand in terms of "u."
  • Changing the differential (\(dx\) or \(du\)) accordingly.
For the given exercise, the substitution \(u = 3\sqrt{x}\) was made, allowing us to reframe the original complex integral into a more manageable form. By substituting the expressions for \(x\) and \(dx\) with those in terms of "u," following through with the derivative aids in the reformulation of the integral. After simplifying, new bounds for definite integrals can also be determined if needed.
Diving into Definite Integrals
Definite integrals calculate the accumulation of a quantity, typically the area under a curve. They are evaluated between two specific limits, unlike indefinite integrals, which lack such boundaries. This results in a definite number rather than a general function with \(C\) added.

To evaluate a definite integral with substitution, remember:
  • Change the integration limits to reflect the new variable "u."
  • Ensure that you incorporate the bounds in your substitution equations.
  • Evaluate the integral and revert to the original variable before computing the numerical result.
In this context, although our task focused directly on integration techniques themselves rather than specific bounds, understanding how to apply substitution can simplify even definite integrals.
Using Integral Tables
Integral tables serve as valuable resources for quickly finding antiderivatives of common forms. They provide readily available solutions for integrals that might otherwise involve complicated algebraic manipulations.

These tables are especially handy when:
  • The form of the integral matches a clean and recognizable pattern from the table.
  • You are dealing with complex integrals that even after substitution seem challenging.
In the exercise provided, once the integral was simplified to \(\int \frac{6}{u^2 + 4}\ du\), an integral table could immediately provide the form for evaluation, leveraging the arctangent function for antiderivatives of \(\int \frac{1}{x^2 + a^2}\ dx\). Thus, they streamline the process by circumventing manual integration steps.
Leverage Computer Algebra Systems
Computer Algebra Systems (CAS) are powerful tools in handling complex mathematical computations. They efficiently evaluate integrals, among other tasks, offering precise results almost instantly. Examples of commonly used CAS include software like Mathematica, Maple, and online tools such as Wolfram Alpha.

Utilizing a CAS comes with benefits:
  • Quick verification of hand-derived solutions.
  • Time-saving for lengthy or intricate calculations.
  • Access to precise and consistent results every time.
In our example, using a CAS confirmed the manually calculated result of \(3 \tan^{-1}\left(\frac{3\sqrt{x}}{2}\right) + C\), ensuring accuracy and providing confidence in the hand-computed solution. CAS tools complement human calculation, highlighting both the ability to cross-verify results and save valuable time.

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Most popular questions from this chapter

Find the volume of the solid generated when the region enclosed by \(y=x^{2} /\left(9-x^{2}\right), y=0, x=0,\) and \(x=2\) is revolved about the \(x\) -axis.

Use any method to find the volume of the solid generated when the region enclosed by the curves is revolved about the \(y\)-axis. $$y=\cos x, y=0, x=0, x=\pi / 2$$

Evaluate the integral by making a substitution that converts the integrand to a rational function. $$\int \frac{\cos \theta}{\sin ^{2} \theta+4 \sin \theta-5} d \theta$$

Use any method to find the volume of the solid generated when the region enclosed by the curves is revolved about the \(y\)-axis. $$y=\ln x, y=0, x=5$$

Let \(f(x)=\cos \left(x^{2}\right)\) (a) Use a CAS to approximate the maximum value of \(\left|f^{\prime \prime}(x)\right|\) on the interval [0,1]. (b) How large must \(n\) be in the midpoint approximation of \(\int_{0}^{1} f(x) d x\) to ensure that the absolute error is less than \(5 \times 10^{-4} ?\) Compare your result with that obtained in Example 6. (c) Evaluate the integral using the midpoint approximation with the value of \(n\) obtained in part (b).
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