Integration - Simplifcation help

[Click2]

Hi all,

Hope someone will be able to help me, I've been sitting at that issue for a bit and I just can't get round it.
I have to simplify the equations before I can preform integration by parts BUT the biggest issue for me is.. to simplify it in a way where I can actually use integration.
I just lose it when there are fractions full of everything (anyone can recommend any good resources to look at?).


I'm just unsure how to treat the roots and powers together - I simplify it more or less but it doesn't seem like my end result is right, it can't be right if I can't integrate it (or at least I don't think I can)
Any explanations how to deal with these, even on different examples would be really helpful.

Thank you,
Click

> edit
forgot to mention - the question asks to simplify and integrate - I think I will need to use integration by parts but I'm stuck at properly simplifying it. The equations used were made up by me - so they might not make much sense - I didn't wanted to ask the same question as my homework since I want to resolve it myself hopefully if I get through the simplifying it to make it work.

 

[HallsofIvy]

A few things you need to know to do these problems:
First, \(\displaystyle \sqrt[n]{x}= x^{1/n}\). Second, the "laws of exponents", \(\displaystyle (x^a)(x^b)= x^{a+b}\) and \(\displaystyle (x^a)^b=x^{ab}\).

For the first problem, \(\displaystyle \frac{12x}{\sqrt[5]{x}\sqrt[3]{x}}\), the denominator is \(\displaystyle (x^{1/5})(x^{1/3})=x^{1/5+ 1/3}= x^{3/15+ 5/15}= x^{8/15}\) so \(\displaystyle \frac{12x}{\sqrt[5]{x}\sqrt[3]{x}}= 12x^{1- 8/15}= 12x^{(15- 8)/15}=12x^{7/15}\).

You did not combine the "\(\displaystyle x\)" and "\(\displaystyle x^{-8/15}\)".
(Also it is never a good idea to use "\(\displaystyle \times\)" as a multiplication symbol when you are using "x" as a variable! Use parentheses instead.)

For the second one, the parentheses need to be around the "\(\displaystyle 11x^2+ 2x\). Also the twentieth root, or 1/20 power, applies to the "2" as well as the exponential. And you need to combine the final powers of x. The denominator is \(\displaystyle \sqrt[20]{2e^{-10x}}= 2^{1/20}e^{-\frac{10x}{20}}= 2^{1/20}e^{-x/2}\) so \(\displaystyle \frac{11x^3+ 2x}{\sqrt{2e^{-10x}}}=\)\(\displaystyle (11x^3+ 2x)2^{-1/20}e^{-x/2}\). You can also write that as \(\displaystyle 2^{-1/20}(11x^3e^{-x/2}+2xe^{-x/2})\).[/tex]

 

[Click2]

Thank you! That was helpful. I knew the rules I just couldn't get my head round how to use them.