Finding the value of b so int_{1}^{b}(x-2)^{3}dx=0

Max Macias

Max Macias

Answered question

2022-08-06

Finding the value of b so 1 b ( x 2 ) 3 d x = 0.
Please how do find b > 1 so that 1 b ( x 2 ) 3   d x = 0 ?
This question is on a chapter dealing with antiderivatives and I'm not sure how to go about it. At this point it is assumed that I don't know how to integrate yet. I'm also not allowed to use the fundamental theorem of calculus.

Answer & Explanation

Izabella Fisher

Izabella Fisher

Beginner2022-08-07Added 14 answers

Step 1
Note that the substitution u = x 2 gives the integral 1 b 2 u 3 d u .
What do you know about integrals of odd functions (in particular what sort will give an integral of 0)?
Step 2
The fact that u u 3 is strictly monotone is enough to show uniqueness of the value thus determined. (Thanks to Fly by Night for pointing out that something additional was required to prove uniqueness.)

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