How Do You Integrate Functions with Polynomial and Rational Powers?

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Discussion Overview

The discussion revolves around techniques for integrating functions with polynomial and rational powers, specifically focusing on the integral of the form {t^{k+n}}/{(1+qt)(1+t)^{2k+3}}dt. Participants explore various substitution methods and integration techniques, including integration by parts and partial fractions.

Discussion Character

  • Technical explanation
  • Mathematical reasoning
  • Homework-related

Main Points Raised

  • One participant asks how to solve the integral {t^{k+n}}/{(1+qt)(1+t)^{2k+3}}dt, noting that n is a natural number.
  • Another participant suggests using the substitution u = 1+t to simplify the integral into a sum of terms that can be handled with integration by parts.
  • A participant attempts to apply the substitution and reformulates the integral but expresses difficulty in solving it using integration by parts.
  • There is a correction where a participant suggests using partial fractions instead of integration by parts, questioning if this method would be easier.
  • Another participant expresses challenges with breaking the integral into partial fractions and requests alternative methods for solving it.
  • A separate integral, t^{k+1}/(1+qt)^{2k+2}dt, is introduced for discussion, specifically over the interval from 0 to 1.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the best method for solving the integrals, with multiple approaches being proposed and challenges noted. The discussion remains unresolved regarding the most effective technique.

Contextual Notes

Participants express uncertainty about the effectiveness of different integration techniques, and there are unresolved steps in the mathematical processes discussed.

Who May Find This Useful

Readers interested in advanced integration techniques, particularly those involving polynomial and rational functions, may find this discussion relevant.

raghavendar24
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Hi how to solve this type of integrals

{t^{k+n}}/{(1+qt)(1+t)^{2k+3}}dt

here n is natural number if some one know how to solve it without n also it is okay.
 
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Hi raghavendar24! :smile:

(try using the X2 tag just above the Reply box :wink:)

The easiest way is probably to substitute u = 1+t, so that it becomes a sum of terms like 1/(a + bu)ur,

and then use integration by parts on each term. :smile:
 
Hi, thanks for reply,


yeah if we substitute the transformation 1+u=t,

the integral tourns out as

Integrate[(u-1)^{k+n}/(1-q+bq)u^{2k+3},{u,1,2}]

i am unable to solve it once again just using integral by parts
 
oops!

raghavendar24 said:
Integrate[(u-1)^{k+n}/(1-q+bq)u^{2k+3},{u,1,2}]

i am unable to solve it once again just using integral by parts
uhh? :confused: oh-oooh :redface: …
tiny-tim said:
The easiest way is probably to substitute u = 1+t, so that it becomes a sum of terms like 1/(a + bu)ur,

and then use integration by parts on each term. :smile:

oops! sorry! I meant use partial fractions. :blushing:

Is that easier? :smile:​
 
Hie,


unable to break it through partail fractions, so can i get any alternative idea to solve it
 
How to solve the integral


t^{k+1}/(1+qt)^{2k+2}dt, t from 0 to 1
 

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