Q. 18 Express each improper integral i... [FREE SOLUTION]

Chapter 5: Q. 18 (page 477)

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

Short Answer

Expert verified

The integral is,

$\lim_{A 鈫� {(\frac{1}{3})}^{-}} {鈭�}_{0}^{A} \frac{1}{3 x^{2} - 4 x + 1} d x + \lim_{B 鈫� {(\frac{1}{3})}^{+}} {鈭�}_{B}^{\frac{1}{2}} \frac{1}{3 x^{2} - 4 x + 1} d x + \lim_{C 鈫� 1^{-}}^{\frac{1}{2}} {鈭�}_{\frac{1}{2}}^{c} \frac{1}{3 x^{2} - 4 x + 1} d x + \lim_{D 鈫� 1^{+}} {鈭�}_{1}^{4} \frac{1}{3 x^{2} - 4 x + 1} d x$

Step by step solution

Step 1. Given information.

We are given an integrals,

${鈭�}_{0}^{4} \frac{1}{3 x^{2} - 4 x + 1} d x$

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 $x 鈫� 鈭�$

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,

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

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91影视

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

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Graph of function

Step 3. Expressing the Integral.

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91影视

Express each improper integral in Exercises15鈥�20 as a sum of limits of proper definite integrals. Do not calculate any integrals or limits; just write them down.鈭�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

Decision Maths

Probability and Statistics

Discrete Mathematics

Calculus

Logic and Functions

Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.

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