Integrating \frac{x^3}{(x+1)^8}: Tips & Hints

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In summary: But I'm grateful for your help nonetheless.In summary, this person has tried to solve a homework problem for a class they're no longer taking, but they don't know where to start. They think they could do partial fractions, but it's 8 variables and 8 integrals. They're grateful for help with the problem from someone else.
  • #1
bob1182006
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Homework Statement


[tex]\int\frac{x^3 dx}{(x+1)^8}[/tex]

Homework Equations


None I can think of.

The Attempt at a Solution


Hm..I just don't know where to start.

No substitution seems to be useful. (u=x+1, u=(x+1)^8).
I've tried pulling x out of the denominator to get:
[tex]\int\frac{x^3 dx}{x^8 (1+x^-1)^8}[/tex]
but that doesn't help either..

This isn't really HW just one of 15 integrals I was given that I should be able to do...

Thanks in Advance for any hints on solving this.
 
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  • #2
bob1182006 said:

Homework Statement


[tex]\int\frac{x^3 dx}{(x+1)^8}[/tex]


Homework Equations


None I can think of.


The Attempt at a Solution


Hm..I just don't know where to start.

No substitution seems to be useful. (u=x+1, u=(x+1)^8).
I've tried pulling x out of the denominator to get:
[tex]\int\frac{x^3 dx}{x^8 (1+x^-1)^8}[/tex]
but that doesn't help either..

This isn't really HW just one of 15 integrals I was given that I should be able to do...

Thanks in Advance for any hints on solving this.
Cal 2? b/c i can't even go at it, I've only completed Cal 1.
 
  • #3
Yep, Calc 2.

I "think" I could do partial fractions but...8 variables to find? and then 8 integrals of increasing power on the denominator. I think there should be an easier way...
 
  • #4
bob1182006 said:
Yep, Calc 2.

I "think" I could do partial fractions but...8 variables to find? and then 8 integrals of increasing power on the denominator. I think there should be an easier way...
man i can't wait to be able to solve these types of problems :-]

i've missed it so much that i even did the even problems that weren't assigned for hw
 
  • #5
u=x+1 is fine. Turns it into (u-1)^3/u^8. Now you just have to expand (u-1)^3. And if that's the worst of your 15 problems, feel blessed. It's really a calc 1 problem.
 
Last edited:
  • #6
Wow thanks, this is about the 2nd time I've ever seen that type of substitution being used >.<.

Yea the problem's weren't really hard Calc 2 stuff mainly things we should know off the top of our heads.
 

FAQ: Integrating \frac{x^3}{(x+1)^8}: Tips & Hints

How do I simplify the expression x3/(x+1)8?

To simplify this expression, you can use the power rule for exponents. First, expand the denominator using the binomial theorem. Then, use the power rule to rewrite the expression as x3-8/(1+8x). Finally, simplify the exponent and combine like terms to get the final expression.

What is the general strategy for integrating x3/(x+1)8?

The general strategy for integrating this expression is to use the substitution method. Let u = x+1, then du = dx. Rewrite the expression as (u-1)3/u8 and substitute u and du into the integral. This will result in a simpler expression that can be integrated using the power rule.

Can I use partial fractions to integrate x3/(x+1)8?

Yes, you can use partial fractions to integrate this expression. First, factor the denominator into its irreducible factors. Then, use the partial fraction decomposition method to write the expression as a sum of simpler fractions. This will allow you to integrate each term separately.

What are some common mistakes to avoid when integrating x3/(x+1)8?

One common mistake to avoid is forgetting to use the substitution method when integrating this expression. It may also be easy to make a mistake when expanding the denominator using the binomial theorem, so be careful with your calculations. Additionally, make sure to simplify the expression as much as possible before integrating to avoid unnecessary complexity.

Can I use software or calculators to integrate x3/(x+1)8?

Yes, you can use software or calculators with integration capabilities to solve this expression. However, it is important to understand the underlying concepts and steps involved in the integration process. Relying solely on technology can hinder your understanding and problem-solving skills.

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