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# Nooby integration question

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1. Say we had the integral of (1+x)1/2, this would be equal to 2/3*(1+x)3/2. However, why can't we do the same tactic with the integral of (1+x2)1/2? For the record, I know how to do it. You use a substitution with x=sinhu. However my question is why can't we add one to the power then divide by the 2x and 1.5? So my question is why doesn't it equal
(1+x2)1.5*2/3*1/(2x)=1/(3x)*(1+x2)1.5?
2. (Original post by jacobe)
Say we had the integral of (1+x)1/2, this would be equal to 2/3*(1+x)3/2. However, why can't we do the same tactic with the integral of (1+x2)1/2? For the record, I know how to do it. You use a substitution with x=sinhu. However my question is why can't we add one to the power then divide by the 2x and 1.5? So my question is why doesn't it equal
(1+x2)1.5*2/3*1/(2x)=1/(3x)*(1+x2)1.5?
You can only integrate in that manner because . It's the reverse chain rule.

. Note the x at the front of the bracket.
As it's not present in , it is not of that form so another technique (substitution as you mention), is required.
3. (Original post by jacobe)
Say we had the integral of (1+x)1/2, this would be equal to 2/3*(1+x)3/2. However, why can't we do the same tactic with the integral of (1+x2)1/2? For the record, I know how to do it. You use a substitution with x=sinhu. However my question is why can't we add one to the power then divide by the 2x and 1.5? So my question is why doesn't it equal
(1+x2)1.5*2/3*1/(2x)=1/(3x)*(1+x2)1.5?
KTA has already said most of it, so:

In general you can integrate in the manner you suggest. This works for and since and your integral is in the form 1 * (1+x)^{1/2} i.e: f'(x)f(x)^{n}.

So, if you had that would also be fine to integrate. But you're missing the essential factor in your integral that disallows the use of the "reverse chain rule".

Also, for a bit of intuition: differentiation is easy (squeezing toothpaste out of the tube) and integration is hard (trying to put the toothpaste back in) so it's not a surprise that you don't have fast-and-hard rules for integration as you do for differentiation.

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