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Chapter 5: Q. 17 (page 464)

Why doesn鈥檛 the definite integral鈭�231-x2dx make sense? (Hint: Think about domains.)

Short Answer

Expert verified

Ans: 1-x2is not defined in [2,3].

Step by step solution

01

Step 1. Given information:

鈭�231-x2dx

02

Step 2. Proving: 

The definite integral 鈭�231-x2 does not make sense because the function has a domain in [-1,1]. But the limit of the integral is [2,3] which does not come under the domain of the function. That's why 1-x2 is not defined in [2,3].

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Most popular questions from this chapter

Solve given definite integral.

鈭�1/41/2鈥�1x21鈭�x2dx

True/False: Determinewhethereachofthestatementsthat follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample.

(a) True or False: f(x)=x+1x-1is a proper rational function.

(b) True or False: Every improper rational function can be expressed as the sum of a polynomial and a proper rational function.

(c) True or False: After polynomial long division of p(x) by q(x), the remainder r(x) has a degree strictly less than the degree of q(x).

(d) True or False: Polynomial long division can be used to divide two polynomials of the same degree.

(e) True or False: If a rational function is improper, then polynomial long division must be applied before using the method of partial fractions.

(f) True or False: The partial-fraction decomposition of x2+1x2(x-3)is of the form Ax2+Bx-3

(g) True or False: The partial-fraction decomposition of x2+1x2(x-3)is of the form Bx+Cx2+Ax-3.

(h) True or False: Every quadratic function can be written in the formA(x-k)2+C

Find three integrals in Exercises 39鈥�74 that can be solved by using a trigonometric substitution of the form x=atanu.

Solve鈭�x4+x2dxthe following two ways:

(a) with the substitution u=4+x2;

(b) with the trigonometric substitution x = 2 tan u.

Find three integrals in Exercises 21鈥�70 that we can anti-differentiate immediately after algebraic simplification.
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