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Prove each statement in Exercises 78–80, 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 abf(x)dx diverges and 0 ≤ f(x) ≤ g(x) for all x ∈ (a, b], thenabg(x)dxalso diverges.

Short Answer

Expert verified

The given statement is proved.

Step by step solution

01

Step 1. Given Information.

The given integrals areabf(x)dxandabg(x)dx.

02

Step 2. Prove.

To prove the given statement integrate each part of inequality and 0 ≤ f(x) ≤ g(x)fromatobwithrespecttox.

So,

=ab0dxabf(x)dxabg(x)dx=0abf(x)dxabg(x)dx

From the inequality, we can depict that abg(x)dxabf(x)dx.It is given that abf(x)dxdiverges so if it diverges then abg(x)dxalso diverges.

Hence, it is proved.

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