Integrate x^3/2 divided by expression - using partial fractions perhaps

In summary, the problem is to solve for a constant in the integral ∫ (x/a)1/2*(x/(x-a)) dx, which can be simplified by using the change of variable u=(x/a)1/2. After simplifying and substituting u, the integral becomes 2/3(x3/a)1/2 + 2(ax)1/2 + aln( ((x/a)1/2-1)/((x/a)1/2+1) ), which is the required answer.
  • #1
Mustaq
2
0

Homework Statement


Hi. My first post!
I'm trying to solve for where a is a constant:

∫ (x/a)1/2*(x/(x-a)) dx


Homework Equations


See above


The Attempt at a Solution


I've tried integration by parts by setting u=(x/a)1/2 but I end up having to solve ∫ (x/a)1/2ln(x-a) - which I can't solve.
I've tried switching them around u=x/(x-a) and end up having to solve ∫x3/2/(x-a)2 - which I can't solve either.
I've thought of using partial fractions but run into x3/2/(x-a)2 which I can't do either.

Any help gratefully received.
Thanks
 
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  • #2
The change of variable u=(x/a)1/2 is very much solvable. Try it again.
 
  • #3
So obvious. I was looking for something really complicated.
Let u = (x/a)1/2
So x=au2, dx/du=2au

First simplify:
∫(x/a)1/2(x/(x-a)) dx
∫(x/a)1/2( 1 + a/(x-a) ) dx

Substitute by u:
∫u( 1 + a/(au2-a) ) 2au du
2a∫u2( 1 + 1/(u2-1) ) du
2/3au3 + 2a∫u2/(u2-1) du
2/3au3 + 2a∫(1 + 1/(u2-1)) du
2/3au3 + 2au + aln( (u-1)/(u+1) )

Finally put back x and get required anaswer:
2/3(x3/a)1/2 + 2(ax)1/2 + aln( ((x/a)1/2-1)/((x/a)1/2+1) )

Thanks, for the hint which lead to the solution (was on it for weeks LOL)
 
Last edited:

1. What is partial fractions?

Partial fractions is a method used to decompose a rational function into simpler fractions. This method is especially useful when integrating rational functions.

2. How do you use partial fractions to integrate x^3/2 divided by expression?

To integrate x^3/2 divided by expression using partial fractions, we first decompose the expression into simpler fractions using the method of partial fractions. Then, we integrate each individual fraction separately.

3. What is the purpose of using partial fractions when integrating?

The purpose of using partial fractions when integrating is to simplify the integration process. It allows us to break down a complex rational function into simpler fractions, making it easier to integrate each fraction separately.

4. Can you provide an example of using partial fractions to integrate x^3/2 divided by expression?

For example, if we have the expression (x^3 + 2x + 1) / (x^2 + x), we can use partial fractions to decompose it into (x + 1) + (1 / x), making it easier to integrate each term separately.

5. Are there any limitations to using partial fractions when integrating?

Yes, there are some limitations to using partial fractions when integrating. This method can only be applied to rational functions, and it may not always be possible to decompose the function into simpler fractions. In some cases, the decomposition may result in complex fractions that are difficult to integrate.

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