SOLUTION: 1) if int(xf(2x+3))dx=10 0n[3,6] find value of int((2x+4)f(2x+7))dx on [1,4] 2) if g(x) is continuous on [-a,a] prove that int(g(x))dx on [-a, a] =int(g(x)+g(-x))dx on[-a, a]

Algebra ->  Graphs -> SOLUTION: 1) if int(xf(2x+3))dx=10 0n[3,6] find value of int((2x+4)f(2x+7))dx on [1,4] 2) if g(x) is continuous on [-a,a] prove that int(g(x))dx on [-a, a] =int(g(x)+g(-x))dx on[-a, a]       Log On


   

Question 1123284: 1) if int(xf(2x+3))dx=10 0n[3,6] find value of int((2x+4)f(2x+7))dx on [1,4]
2) if g(x) is continuous on [-a,a] prove that int(g(x))dx on [-a, a] =int(g(x)+g(-x))dx on[-a, a]
Answer by solver91311(24713) About Me  (Show Source):
You can put this solution on YOUR website!


2. If is continuous on , prove that



Can't be done because the assertion is not true in general:

Counterexample

Assume is an even function that is continuous on some closed interval . By the definition of an even function: . Therefore:

.

This number is twice as large as and therefore not equal to:



I leave it as an exercise for the student to #1, prove that the assertion is false for odd functions as well (Hint: begin with the definition of an odd function) and #2, find a function for which the assertion is true.


John

My calculator said it, I believe it, that settles it