Calculus

10th Edition
Ron Larson + 1 other
ISBN: 9781285057095

Chapter
Section

Calculus

10th Edition
Ron Larson + 1 other
ISBN: 9781285057095
Textbook Problem

Making an Integral Improper For each integral, find a nonnegative real number b that makes the integral improper. Explain your reasoning. ∫ b 0     1 x 2   − 9   d x ∫ b 0     1 4 − x   d x ∫ b 0     x x 2   − 7 x   +   12     d x ∫ 10 b     ln   x   d x ∫ b 0     tan   2 x   d x ∫ b 0   cos x 1 − sin x   d x

(a)

To determine

The nonnegative real number b that makes the improper integrals for 0b1x29dx.

Explanation

There are two different types of improper integral.

The first improper integral is;

1. Integrals over the infinite interval:

The improper integrals wherein one or both of the limits of the integrals is/are infinity are called as integrals over an infinite interval. One or both of the limits is/are ± in integrals over an infinite interval.

The second improper integral is;

2. Integrals with discontinuous integrand.

The improper integrals wherein either one or both limits of these integrals or the points between them is/are points of infinite discontinuity are called integrals with discontinuous integrand.

Hence, the two kinds of integrals are integrals over infinite interval and integrals with discontinuous integrand.

Example:

Consider the integral 0xdx.

Then, the above integral is improper integral, as it is integral over an infinite interval.

The second example is 021x1dx.

In the above integral, the function 1x1 has discontinuity at x=1.

So, the integral 021x1dx is improper integral

(b)

To determine

The nonnegative real number b that makes the improper integrals for 0b14xdx.

(c)

To determine

The nonnegative real number b that makes the improper integrals for 0b1x27x+12dx.

(d)

To determine

The nonnegative real number b that makes the improper integrals for b10lnxdx.

(e)

To determine

The nonnegative real number b that makes the improper integrals for 0btan2xdx.

(f)

To determine

The nonnegative real number b that makes the improper integrals for 0bcosx1sinxdx.

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