Chapter 5: Q. 18 (page 477)

Express each improper integral in Exercises15–20 as a sum of limits of proper definite integrals. Do not calculate any integrals or limits; just write them down.
$\int_{0}^{4} \frac{1}{3 x^{2} - 4 x + 1} d x$

Short Answer

Expert verified

The integral is,

$\lim_{A \to {(\frac{1}{3})}^{-}} \int_{0}^{A} \frac{1}{3 x^{2} - 4 x + 1} d x + \lim_{B \to {(\frac{1}{3})}^{+}} \int_{B}^{\frac{1}{2}} \frac{1}{3 x^{2} - 4 x + 1} d x + \lim_{C \to 1^{-}}^{\frac{1}{2}} \int_{\frac{1}{2}}^{c} \frac{1}{3 x^{2} - 4 x + 1} d x + \lim_{D \to 1^{+}} \int_{1}^{4} \frac{1}{3 x^{2} - 4 x + 1} d x$

Step by step solution

Step 1. Given information.

We are given an integrals,

$\int_{0}^{4} \frac{1}{3 x^{2} - 4 x + 1} d x$

Step 2. Graph of function

Let $f (x) = \frac{1}{3 x^{2} - 4 x + 1}$ ,

The graph is as follows,

Step 3. Expressing the Integral.

It can be seen from the graph that the function has vertical asymptotes at $x = \frac{1}{3}$ and $x = 1$ Interval of integration contains points $\frac{1}{3}, 1$ , which are not in the domain.

That is split the integral at some points, $x = \frac{1}{3}, x = 1$ , in order to separately consider the limit as $x \to \infty$

So, divide the interval in the following subintervals $(0, \frac{1}{3}), (\frac{1}{3}, \frac{1}{2}), (\frac{1}{2}, 1), (1, 4)$ .

So,

$\int_{0}^{4} \frac{1}{3 x^{2} - 4 x + 1} d x = \lim_{A \to {(\frac{1}{3})}^{-}} \int_{0}^{A} \frac{1}{3 x^{2} - 4 x + 1} d x + \lim_{B \to {(\frac{1}{3})}^{+}} \int_{B}^{\frac{1}{2}} \frac{1}{3 x^{2} - 4 x + 1} d x + \lim_{C \to 1^{-}}^{\frac{1}{2}} \int_{\frac{1}{2}}^{c} \frac{1}{3 x^{2} - 4 x + 1} d x + \lim_{D \to 1^{+}} \int_{1}^{4} \frac{1}{3 x^{2} - 4 x + 1} d x$

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.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Express each improper integral in Exercises15–20 as a sum of limits of proper definite integrals. Do not calculate any integrals or limits; just write them down.
$\int_{0}^{4} \frac{1}{3 x^{2} - 4 x + 1} d x$

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Graph of function

Step 3. Expressing the Integral.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Decision Maths

Discrete Mathematics

Calculus

Probability and Statistics

Geometry

Study anywhere. Anytime. Across all devices.

Express each improper integral in Exercises15–20 as a sum of limits of proper definite integrals. Do not calculate any integrals or limits; just write them down.∫0413x2-4x+1dx

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Graph of function

Step 3. Expressing the Integral.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Decision Maths

Discrete Mathematics

Calculus

Probability and Statistics

Geometry

Study anywhere. Anytime. Across all devices.

Express each improper integral in Exercises15–20 as a sum of limits of proper definite integrals. Do not calculate any integrals or limits; just write them down.
$\int_{0}^{4} \frac{1}{3 x^{2} - 4 x + 1} d x$