(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

By appealing to geometric evidence show that

[tex]\int_0^8x^n\,dx + \int_0^1 x^{1/n}\,dx = 1[/tex]

for n a positive integer.

2. Relevant equations

Fundamental theorem of calculus, power rule for integration.

3. The attempt at a solution

I integrated. For the first integral, I get:

[tex] \frac{8^{n+1}}{n+1}[/tex]

and for the second:

[tex] \frac{n}{n+1} [/tex]

As an experiment, I tried this for a few values of n, for example, n=1 gives [itex] 8^2/2 + 1/2 = 32.5 [/tex], which is certainly not 1.

So obviously something is awry here. I think I integrated properly, but perhaps not.

Can someone shed some light on this for me?

Thank you,

Dorothy

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# Definite Integral Problem

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