Integration indices problem

1. Mar 6, 2008

_Mayday_

I have a bit of a problem with this question, I will do my best to offer an answer. I think the problem is not with the differentiation but with my indices.

Here is the initial formula.

$$\int (6x + 2 + x^{-\frac{1}{2}}) dx$$

Here is my attempt

$$\frac{6x^2}{2} + 2x + \frac{x\frac{1}{2}}{\frac{1}{2}}$$

$$3x^2 + 2x + \frac{\sqrt\frac{1}{2}}{\frac{1}{2}} + c$$

Where have I gone wrong? I'll bet it's the whole indices thing!

Thanks for any help

2. Mar 6, 2008

Tedjn

Do you mean:

$$\frac{\sqrt{x}}{\frac{1}{2}} = 2\sqrt{x}$$

I don't see how the x disappeared in the last term of your solution.

3. Mar 6, 2008

dynamicsolo

The problem is in the last term: that should be

$$\frac{x^{\frac{1}{2}}}{\frac{1}{2}}$$ ,

which is to say, x to the one-half power divided by one-half. X to the one-half power is the square root of x, but in any case, what you did in the next line was to drop the "x". You want to say

$$\frac{x^{\frac{1}{2}}}{\frac{1}{2}} = 2 {x^{\frac{1}{2}}$$ .

4. Mar 6, 2008

_Mayday_

Ah ok, why does it do that then? I don't understand that could you take me through it please.

Thanks to both of you so far

5. Mar 6, 2008

Tedjn

Do you mean, why is dividing by 1/2 equal to multiplying by 2?

6. Mar 6, 2008

_Mayday_

Well, how does this work?

$$\frac{x^{\frac{1}{2}}}{\frac{1}{2}} = 2 {x^{\frac{1}{2}}$$ .

7. Mar 6, 2008

dynamicsolo

You want to "eliminate" the 1/2 in the denominator of this quotient, which means making it 1, so that you end up with the numerator divided by 1, which is just the numerator. You can multiply the numerator and denominator by 2, which is like multiplying your quotient by 2/2 = 1, which means the value of the fraction is unchanged. You will end up with

$$\frac{2x^{\frac{1}{2}}}{1} = 2 {x^{\frac{1}{2}} = 2\sqrt{x}$$

Last edited: Mar 6, 2008
8. Mar 6, 2008

_Mayday_

Brilliant! Thanks for your time you two! Funny how trivial these things seem in hind sight! :shy:

Thanks Again!

9. Mar 6, 2008

_Mayday_

So then my final equation will be:

$$3x^2 + 2x + 2x^{\frac{1}{2}} + c$$

10. Mar 6, 2008

dynamicsolo

If you want to check it, differentiate it term-by-term: you'll get your original integrand back again... (It's correct!)