/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 6 Use the Table of Integrals on Re... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 6

Use the Table of Integrals on Reference Pages \(6-10\) to evaluate the integral. $$\int_{2}^{3} \frac{1}{x^{2} \sqrt{4 x^{2}-7}} d x$$

Short Answer

Expert verified
Use a suitable substitution and table integral to evaluate; it involves inverse trigonometric functions and definite calculation.

Step by step solution

01

Identify the Formula

Look for a suitable formula from the Table of Integrals. We need a formula that matches the pattern of our integral's function.
02

Match the Integral to the Formula

Check if the function matches a form like \( \int \frac{1}{u^2 \sqrt{a^2 u^2 - b}} \, du \). After examining, we find the case \( \int \frac{1}{x^2 \sqrt{4x^2 - 7}} \, dx \) matches the integrals that transform expressions involving square roots.
03

Perform Substitution

Perform a trigonometric substitution if necessary. Set \( x = \frac{a}{b} \sec(\theta) \) where such functions simplify to \( \int \sec^{-1}(\theta) \, d\theta \). Since \( a = \sqrt{4} \) and \( b = \sqrt{7} \), we focus on substituting for \( \sqrt{4x^2 - 7} \).
04

Solve the Integral

Use the integral formula identified and the correct substitution to solve the integral, which will result in an inverse trigonometric function or a simplified algebraic form. There might be integration by parts if necessary.
05

Evaluate the Definite Integral

Now with the solved indefinite integral expression and after reversing any substitutions, evaluate the integral from \( x = 2 \) to \( x = 3 \).
06

Calculate the Integral

Plug in the upper limit and lower limit into the evaluated expression and solve for the final result.

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 91影视!

Key Concepts

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

Trigonometric Substitution in Integration
When dealing with complex integrals, especially those involving square roots like \( \int \frac{1}{x^2 \sqrt{4x^2 - 7}} \, dx \), trigonometric substitution can simplify the process. This technique involves using a trigonometric function to substitute for a part of the integral. Here's how it usually works:
  • You identify a substitution that will transform part of the integral into a simpler form, often using \( \sin(\theta) \), \( \cos(\theta) \), or \( \tan(\theta) \).
  • The goal is to eliminate the square root and make the integral more manageable.
  • In cases involving expressions like \( a^2 - b^2x^2 \), substituting \( x = \frac{a}{b} \sec(\theta) \) is useful because \( \sec(\theta) \) relates closely to expressions under square roots.
For our problem, setting x equal to a trigonometric function of \(\theta\) can help unravel the complex expression \( \sqrt{4x^2 - 7} \). Substitution rewrites the integral in terms of \(\theta\), making it easier to solve. Once rewritten, the integrand may resemble a simpler form in the table of integrals.
Inverse Trigonometric Functions
After performing trigonometric substitution, the indefinite integral is often solved using inverse trigonometric functions. These functions, like \( \tan^{-1}(x) \), \( \sin^{-1}(x) \), and \( \cos^{-1}(x) \), provide a way to revert the substitution back to the original variable.
  • Inverse trigonometric functions frequently appear when the substitution involves expressions under square roots.
  • In our earlier example, \( \int \frac{1}{x^2 \sqrt{4x^2 - 7}} \, dx \) transforms through substitution and simplification to involve an inverse function.
  • These functions map a ratio back to angles, effectively undoing the initial substitution.
Solving the integral via this method often means the end result simplifies to these inverse functions. Not only do they allow you to evaluate the integral, but they also ease reversing the substitution and calculating definite integrals.
Integration Techniques
The journey of evaluating a definite integral combines various integration techniques, which are fundamental to solving integrals like \( \int_{2}^{3} \frac{1}{x^{2} \sqrt{4 x^{2}-7}} d x \). These techniques range from substitution methods to transformations using known integral formulas.
  • First, recognize patterns in the integrand that match known forms in the integral table. These are pre-evaluated, making the task easier.
  • In difficult scenarios, use integration techniques like substitution, directly altering the variable or trigonometric form to simplify.
  • After substitution and solving the indefinite integral, employ the Fundamental Theorem of Calculus to evaluate between specific limits (e.g., from 2 to 3).
  • Final steps include substituting back any variable changes to express results in the original variable, then finding the numerical value between set limits.
  • Integration by parts may arise if products of functions are involved, though it's not explicitly necessary in this context.
Through these methods, solving complex integrals becomes manageable, combining theory with strategic application of integration rules. With practice, you can identify the best approach for various integral structures.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

\(7-18\) Use (a) the Trapezoidal Rule, (b) the Midpoint Rule, and (c) Simpson's Rule to approximate the given integral with the specified value of \(n .\) Round your answers to six decimal places.) $$ \int_{0}^{4} \sqrt{1+\sqrt{x}} d x, \quad n=8 $$

Find the value of the constant \(C\) for which the integral $$\int_{0}^{\infty}\left(\frac{x}{x^{2}+1}-\frac{C}{3 x+1}\right) d x$$ converges. Evaluate the integral for this value of \(C\)

Show that if 0\(\leqslant f(t) \leqslant M e^{a \prime}\) for \(t \geqslant 0,\) where \(M\) and \(a\) are constants, then the Laplace transform \(F(s)\) exists for \(s>a\)

\(\begin{array}{l}{62-63 \text { Find the area of the region under the given curve from }} \\ {1 \text { to } 2 \text { . }}\end{array}\) $$ y=\frac{1}{x^{3}+x} $$

Find the escape velocity \(v_{0}\) that is needed to propel a rocket of mass \(m\) out of the gravitational field of a planet with mass \(M\) and radius \(R\) . Use Newton's Law of Gravitation (see Exer- cise 29 in Section 6.4 ) and the fact that the initial kinetic energy of \(\frac{1}{2} m n_{0}^{2}\) supplies the needed work.
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Probability and Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Statistics

Read Explanation

Geometry

Read Explanation

Pure Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.