Chapter 4: Q. 17 (page 362)

Show by exhibiting a counterexample that, in general, $\int f (x) g (x) d x \neq (\int f (x) d x) (\int g (x) d x)$ . In other words, find two functions f and g so that the integral of their product is not equal to the product of their integrals.

Short Answer

Expert verified

The counterexample is $f (x) = x; g (x) = \frac{1}{x} .$

Step by step solution

Step 1. Given information.

The given inequality is $\int f (x) g (x) d x \neq (\int f (x) d x) (\int g (x) d x) .$

Step 2. Conclusion.

Let the two functions be,

$f (x) = x; g (x) = \frac{1}{x}$

Now,

$\int f (x) g (x) d x = \int x \frac{1}{x} d x = x + C (\int f (x) d x) (\int g (x) d x) = = (\int x d x) (\int \frac{1}{x} d x) = (\frac{x^{2}}{2} + C) (\ln x + C^{'}) T h e r e f o r e, \int f (x) g (x) d x \neq (\int f (x) d x) (\int g (x) d x)$

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Show by exhibiting a counterexample that, in general, $\int f (x) g (x) d x \neq (\int f (x) d x) (\int g (x) d x)$ . In other words, find two functions f and g so that the integral of their product is not equal to the product of their integrals.

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Conclusion.

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Show by exhibiting a counterexample that, in general, ∫f(x)g(x)dx≠∫f(x)dx∫g(x)dx. In other words, find two functions f and g so that the integral of their product is not equal to the product of their integrals.

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Conclusion.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

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Study anywhere. Anytime. Across all devices.

Show by exhibiting a counterexample that, in general, $\int f (x) g (x) d x \neq (\int f (x) d x) (\int g (x) d x)$ . In other words, find two functions f and g so that the integral of their product is not equal to the product of their integrals.