Simple questions regarding antiderivative rules and fractions

I'm just having a bit of trouble remembering the process of some of the integral rules, namely the power rule and the 'indefinite integral of a constant multiple of a function.

Basically, the example in my textbook is as follows (Not sure how to do the integral symbol, so hope this all make sense - I'll use 'int.' to denote that symbol).

$\displaystyle int. (1/x^(^3^/^2^)dx = int. x^(^-^3^/^2^)dx$ (No issues understanding this)

$\displaystyle =(1/(-1/2))x^(^-^1^/^2^) + C$ (Again, no issues here)

$\displaystyle = -2x^(^-^1^/^2^) + C = -(2/x^(^1^/^2^)) + C$

This is where I get a little lost. I have no idea where the 2 comes from in this part. I would have thought that from $\displaystyle =(1/(-1/2))x^(^-^1^/^2^)$ it would in fact be $\displaystyle ((-1/2)/(-1/2)x^(^-^1^/^2^)$, which would just be x^(-1/2).

Obviously, I'm going wrong here, so an explanation would be great. The textbook makes no mention of why this occurs and it's most likely something simple..

Cheers

Re: Simple questions regarding antiderivative rules and fractions

Quote:

Originally Posted by

**astuart** I'm just having a bit of trouble remembering the process of some of the integral rules, namely the power rule and the 'indefinite integral of a constant multiple of a function.

Basically, the example in my textbook is as follows (Not sure how to do the integral symbol, so hope this all make sense - I'll use 'int.' to denote that symbol).

$\displaystyle int. (1/x^(^3^/^2^)dx = int. x^(^-^3^/^2^)dx$ (No issues understanding this)

$\displaystyle =(1/(-1/2))x^(^-^1^/^2^) + C$ (Again, no issues here)

$\displaystyle = -2x^(^-^1^/^2^) + C = -(2/x^(^1^/^2^)) + C$

This is where I get a little lost. I have no idea where the 2 comes from in this part. I would have thought that from $\displaystyle =(1/(-1/2))x^(^-^1^/^2^)$ it would in fact be $\displaystyle ((-1/2)/(-1/2)x^(^-^1^/^2^)$, which would just be x^(-1/2).

Obviously, I'm going wrong here, so an explanation would be great. The textbook makes no mention of why this occurs and it's most likely something simple..

Cheers

First, the command for the integral symbol is \int.

Now,

$\displaystyle \displaystyle \begin{align*} \frac{1}{-\frac{1}{2}} &= 1 \div \left(-\frac{1}{2}\right) \\ &= 1 \times \left(-\frac{2}{1}\right) \\ &= -2 \end{align*}$

Re: Simple questions regarding antiderivative rules and fractions

Thanks Prove it.

It appears I wasn't simplifying the fraction to get -2, but was instead copying the n from x^n into the numerator when that wasn't necessary at that stage.

Cheers.