Question

Exercises $5-12,$ decide whether the integral is improper. Explain your reasoning. $$\int_{0}^{1} \frac{2 x-5}{x^{2}-5 x+6} d x$$

Name: Problem 7, Chapter 8: Calculus of a Single Variable
Uploaded: 2020-01-17T03:38:25-08:00
Duration: 2 min 31 s
Channel: Luuk Verhoeven
Description: Exercises 5-12, decide whether the integral is improper. Explain your reasoning. ∫0^1 (2 x-5)/(x^2-5 x+6) d x

   Exercises $5-12,$ decide whether the integral is improper. Explain your reasoning.
$$\int_{0}^{1} \frac{2 x-5}{x^{2}-5 x+6} d x$$

Calculus of a Single Variable

Ron Larson 11th Edition

Chapter 8, Problem 7

Instant Answer

Solved by Expert Luuk Verhoeven

01/17/2020

Step 1

The limits of the integral are 0 and 1, which are both finite numbers. So, the integral is not improper due to infinite limits. Show more…

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Exercises $5-12,$ decide whether the integral is improper. Explain your reasoning. $$\int_{0}^{1} \frac{2 x-5}{x^{2}-5 x+6} d x$$

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Key Concepts

Improper Integral

An improper integral is one in which either the interval of integration is unbounded or the integrand becomes infinite at one or more points within the interval. This requires special techniques, like taking limits, to properly evaluate the integral.

Discontinuity Analysis

When determining if an integral is improper, it is essential to analyze the integrand for any discontinuities, such as points where the denominator is zero in a rational function. Identifying these points helps ascertain whether the integral needs to be treated as improper.

Rational Function Behavior

Rational functions, as ratios of polynomials, can have points where they are undefined if the denominator is zero. In the context of integration, it is important to check whether such points lie within the interval of integration to determine if the function remains continuous throughout the interval.

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