Hi, I'm having a little bit of trouble with a integration question, here is the text from the question.

Find the antiderivitive of the given functions:

Basically I just have no idea how to expand

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- Jan 18th 2011, 01:45 PMCrellIntegral Refresher Question
Hi, I'm having a little bit of trouble with a integration question, here is the text from the question.

Find the antiderivitive of the given functions:

Basically I just have no idea how to expand - Jan 18th 2011, 01:48 PMdwsmith
- Jan 18th 2011, 01:53 PMHallsofIvy
Well, you could expand it using the "binomial theorem":

For

But you**don't**want to do that. Instead, make the substituition so that and . - Jan 18th 2011, 02:20 PMCrell
Thank you much, this is the part I was missing.

Aside:

Let

Derive this equation

Solve for

Manipulate the first equation to make it appear like so that when you substitute dx, the x is eliminated.

Therefore:

Substitute into first equation

Integrate, sub in u and simplfy.

- Jan 18th 2011, 02:26 PMPlato
Did you differentiate that answer to cheek it for yourself?

- Jan 18th 2011, 02:35 PMCrell
Have now, thanks.

- Jan 18th 2011, 02:37 PMQuacky
This is a perfect place to use my favourite rule of integration:

So let

You'll notice that itquite a match to what you are trying to integrate. However, if you divide by , it is*isn't***exactly**what you are trying to integrate. Therefore, we can apply the rule as normal, but we must remember to divide our final answer by -2.

I think in this case, this method is much nicer than substitution - if you spot the rule, it will take a few lines of working to obtain the full solution with no messy work required at all. - Jan 18th 2011, 03:29 PMTheCoffeeMachine
In speaking of the binomial theorem, you can find the anti-derivative

without performing an expansion, but in somewhat undesirable forms:

or .

Plus the constant of integration. The u-sub does the job far better.

__Spoiler__: