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Chapter 4: Q. 16 (page 362)

Show by exhibiting a counterexample that, in general, 鈭�f(x)g(x)dx鈮�鈭�f(x)dx鈭�g(x)dx. In other words, find two functions f and g such that the integral of their quotient is not equal to the quotient of their integrals.

Short Answer

Expert verified

The counterexample isf(x)=x2,g(x)=x.

Step by step solution

01

Step 1. Given Information.

The given inequality to prove is鈭�f(x)g(x)dx鈮�鈭�f(x)dxlg(x)dx.

02

Step 2. Conclusion.

Let the two functions be:

f(x)=x2,g(x)=x

Now, we can write,

鈭�f(x)g(x)dx=鈭�x2xdx=鈭�xdx=x22+C

Also,

localid="1648575504638" 鈭�f(x)dx鈭�g(x)dx==鈭�x2dx鈭�xdxTherefore,鈭�f(x)g(x)dx=x33+Cx22+C鈮�=鈭�f(x)dx鈭�g(x)dx

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Most popular questions from this chapter

Given a simple proof that鈭�k=5n(ak+bk)=鈭�k=5nak+鈭�k=5nbk

Use the Fundamental Theorem of Calculus to find the exact values of the given definite integrals. Use a graph to check your answer.

鈭�01鈥�11+x2dx

Repeat Exercise 13 for the function f shown above at the right, on the interval [-2,2]

Consider the sequence A(1), A(2), A(3),.....,A(n) write our the sequence up to n. What do you notice?

Fill in each of the blanks:

(a) 鈭�dx=x6+C.

(b) x6 is an antiderivative of .

(c) The derivative of x6is .
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