What step am i missing? I don't understand how I can get to 1/5 * 2/3 (x+1)^(3/2) integrated from 3 to 8 from 1/5 * integral from 3 to 8 of (x+1)^(1/2) dx...

Question

anonymous · Answer

$$1/5 \int\limits_{3}^{8}(x+1)^(1/2) $$

anonymous · Answer

to

anonymous · Answer

$$1/5 * 2/3 (x+1) ^ (3/2) integrated from 3 \to 8$$

anonymous · Answer

What's happening here is an application of the rule for integration that deals with functions raised to powers.  The general version is:
$$\int\limits_{a}^{b}x^n = \frac{ 1 }{ n+1 }x^{n+1}$$ from a to b.

In this case, n is 1/2, so plugging that value into the general form yields:
$$(1/5)\int\limits_{3}^{8}x^{1/2}=(1/5)\frac{ 1 }{ 1/2 + 1 }x^{1/2 +1}=(1/5)\frac{ 1 }{ 3/2 }x^{3/2}=(1/5)\frac{ 2 }{ 3 }x^{3/2}$$ from 3 to 8.

anonymous · Answer

but this is (x+1) to the 1/2 power. doesn't that change it?

anonymous · Answer

Ah, sorry I missed that.  It doesn't change much about the answer, just replace x with (x + 1).

There are cases where the difference would change the answer more dramatically, but since x and x+1 have the same derivative, there's no difference here.

anonymous · Answer

ok. thank you. just wondering, could that problem with different derivatives be solved by using the substitution method?

anonymous · Answer

Yeah, that would address that issue perfectly.

anonymous · Answer

thank you again for your help.