Partial fractions with fractional powers

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SUMMARY

The discussion focuses on integrating the function \(\frac{1+x}{(2+x)^{3/2}}\) using partial fractions. Participants clarify that while the integration can be performed, it does not strictly adhere to the traditional method of partial fraction decomposition due to the presence of fractional powers. A substitution method is suggested, specifically letting \(u=(2+x)^{1/2}\), which allows for polynomial decomposition into partial fractions. The final integrated result is confirmed as \(2 \sqrt{x+2} (1+\frac{1}{x+2})\).

PREREQUISITES
  • Understanding of integration techniques, specifically for rational functions.
  • Familiarity with partial fraction decomposition methods.
  • Knowledge of substitution methods in calculus.
  • Basic algebraic manipulation of polynomials and fractional powers.
NEXT STEPS
  • Study the method of substitution in integration, focusing on cases with fractional powers.
  • Learn about partial fraction decomposition in detail, including its limitations.
  • Practice integrating functions that involve both polynomial and fractional components.
  • Explore advanced integration techniques, such as integration by parts and trigonometric substitution.
USEFUL FOR

Students and educators in calculus, particularly those dealing with integration techniques and partial fractions. This discussion is beneficial for anyone looking to deepen their understanding of integrating functions with fractional powers.

jdstokes
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Homework Statement



How does one integrate e.g. \frac{1+x}{(2+x)^{3/2}} by partial fractions?

The Attempt at a Solution



I have no idea about this. I've never seen this technique applied with fractional powers before.
 
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This looks interesting, I have never done it but my guess would be to make a substitution.
 
Hint: x+1=(x+2)-1:wink:
 
Nice catch gabba, can't believe I didnt see that ^_^. Though, does that satisfy it being a partial fraction after you simplify/reduce? (Looks up the def.)
 
Yes, nice one. Is splitting the numerator in this way a special case of partial fractions? Not that it really matters...
 
No. That is NOT using "partial fractions". This can be integrated but not by partial fractions.
 
jdstokes said:

Homework Statement



How does one integrate e.g. \frac{1+x}{(2+x)^{3/2}} by partial fractions?

The Attempt at a Solution



I have no idea about this. I've never seen this technique applied with fractional powers before.

is the ans
2 \sqrt{x+2} (1+\frac{1}{x+2})
 
Last edited:
icystrike said:
is the ans
2 \sqrt{x+2} (1+\frac{1}{x+2})

Yup :smile:
 
HallsofIvy said:
No. That is NOT using "partial fractions". This can be integrated but not by partial fractions.

Sure it is: if you let u=(2+x)^{1/2} then both the numerator and denominator are polynomials in powers of u. You can the decompose it into partial fractions by letting \frac{1+x}{(2+x)^{3/2}}=\frac{u^2-1}{u^3}=A+\frac{B}{u}+\frac{C}{u^2}+\frac{D}{u^3} and determining A,B,C and D...or you can simply recognize that the fraction decomposes into \frac{1}{(2+x)^{1/2}}-\frac{1}{(2+x)^{3/2}}

Either way, it sure seems like "partial fraction decomposition" to me.
 
  • #10
gabbagabbahey said:
Yup :smile:

hahas. thanks loads for your hint :smile:
CHEERS!
 

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