Question

How does one antidifferentiate x*(x^2+1)^(-3/2)?

Thanks in advance,
Nick

Answer 1

if i help u will u help me

Answer 2

try using substitution.

Answer 3

Sure, as long as it is the right answer :)

Answer 4

is there anywhere you see a derivative of some function multiplied by the function itself in the problem?

Answer 5

subsatute every thing and theres ur answer

Answer 6

Lilith, I haven't learned substitution yet, and I want to try without it

Answer 7

Although, I will be soon.

Answer 8

ok. well then you can't antidifferentiate without factoring out the polynomial. so how would you factor out (x^2)^(-3/2) first?

Answer 9

Well I took (x^2)^3, which was x^6 + 3x^4 + 3x^2 + 1. And then I figured trying to take the square root of that might be beneficial. But I've been similarly stuck on this step to :/

Answer 10

i took a second look. because we have (x^2+1) as the base we cannot factor without have imaginary numbers

Answer 11

this means you have to use substitution. i can walk you through it if you would like

Answer 12

I'll give you a best response, but no thank you :) I'm certain there is a way to do this without substitution. Else this 1980's textbook is defective..

Answer 13

ok :) i will get back if i can figure a way out without substitution :)

Answer 14

Sounds good!

Answer 15

Nick there is a method called "Advanced guessing", it allows you to skip doing a substitution if you're able to recognize the term and it's derivative in your problem.

It's based off of intuition and having a foundation of understanding the substitution method. You cannot solve this without first learning the substitution method.

Answer 16

that is what i had thought.

Answer 17

you could also do a taylor series expansion to approximate the function but that, too, will most likely be a bit complicated

Answer 18

Oh boy that would suck XD lol

Answer 19

If it helps, this is a verification problem stating: The integral from a to b of x*(x^2+1)^(-3/2) = -1*(x^2+1) from a to b.

Answer 20

And thanks for all the insight so far!

Answer 21

Woops, it actually says: The integral from a to b of x*(x^2+1)^(-3/2) = -1*(x^2+1)^(-1) from a to b.

Answer 22

FOR REAL, it says: The integral from a to b of x*(x^2+1)^(-3/2) = -1*(x^2+1)^(-1/2). Hence the negative square root at the end.

Answer 23

\[\large -(x^2+1)^{-1/2}\]This should be the answer :D

It sounds like the book is just providing you with the answer, it doesn't give any steps?
But yah, it's based off of letting

u=x^2+1
Then when you take the derivative of your substitution
and you're able to replace some stuff and you'll have a much easier anti-derivative to take at that point.

Answer 24

lol darn you beat me to it ^^

Answer 25

Haha, well, you have credibility! This book is pretty irritating, it's not listing any steps to achieve verification..Maybe I'll cheat and just use substitution for now.

Answer 26

that's what i'd do.

Answer 27

Oh that's cheating? You're suppose to solve the problem without using substitution? Or do you say that because you haven't learned it yet? :D
Sounds like maybe the teacher just assigned that problem by accident XD heh

Answer 28

Yeah, haven't learned it yet. Following this near-vintage 80's book, but homeschooled :)

Answer 29

Oh fun ^^