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Chapter 5: Q. 79 (page 479)

Prove each statement in Exercises 78鈥�80, using the definition of improper integrals as limits of proper definite integrals.

Suppose f(x) and g(x) are both continuous on (a, b] but not at x = a. If 鈭�abf(x)dx converges and 0 鈮� g(x) 鈮� f(x)for all x 鈭� (a, b], then鈭�abg(x)dx also converges.

Short Answer

Expert verified

The given statement is proved.

Step by step solution

01

Step 1. Given Information.

The given integrals are鈭�abf(x)dxand鈭�abg(x)dx.

02

Step 2. Prove.

To prove the given statement integrate each part of inequality and 0 鈮� g(x) 鈮� f(x)from atobwithrespecttox.

So,

=鈭�ab0dx鈮�鈭�abg(x)dx鈮�鈭�abf(x)dx=0鈮�鈭�abg(x)dx鈮�鈭�abf(x)dx

From the inequality, we can depict that 鈭�abf(x)dx鈮�鈭�abg(x)dx.It is given that 鈭�abf(x)dxconverges so if it converges then 鈭�abg(x)dxalso converges.

Hence, it is proved.

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