Evaluating an indefinite integral

In summary, the conversation discusses the integral \int \frac{(a-x)^{r/s-1}}{(b-x)^{r/2}}dx and attempts to find a solution using a substitution. However, it is concluded that there is no regular solution and a numerical approximation would be necessary for practical use.
  • #1
sara_87
763
0

Homework Statement



Evaluate the integral

[itex]\int \frac{(a-x)^{r/s-1}}{(b-x)^{r/2}}dx[/itex]

Homework Equations



given: s>r

The Attempt at a Solution



I tried using a substitution:
let u=b-x
so du=-dx
this gives:

[itex]-\int \frac{(a-b+u)^{r/s-1}}{u^{r/2}}du[/itex]

I don't know what i should do after this, and i think this substitution will take me nowhere.

Does anyone have any ideas?

Thank you in advance
.
 
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  • #2
Is the exponent in the numerator (r/s)-1, or is it r/(s-1) ?
 
  • #3
the exponent is (r/s)-1
 
  • #4
Hi sara_87! :smile:

I believe there is no solution for your integral using only a finite number of standard functions.

If you feed it to WolframAlpha:
http://www.wolframalpha.com/input/?i=\int+\frac{%28a-x%29^{r%2Fs-1}}{%28b-x%29^{r%2F2}}dx
WolframAlpha comes up with a Hypergeometric function F.
With this F your integral can be expressed, but as far as I'm concerned that's just another way of saying it does not have a regular solution.

To use your integral in practice, you'd normally use a numerical approximation.
 

1. What is an indefinite integral?

An indefinite integral is a mathematical concept that refers to the process of finding the antiderivative of a given function. It is represented by the symbol ∫, and is used to calculate the total change or cumulative effect of a continuous change over an interval.

2. How is an indefinite integral evaluated?

An indefinite integral is evaluated using a set of rules and techniques, such as the power rule, substitution, and integration by parts. These methods involve manipulating the given function to simplify it and then applying the appropriate rule to find the antiderivative.

3. What is the difference between an indefinite integral and a definite integral?

The main difference between an indefinite integral and a definite integral is that an indefinite integral produces a general solution, while a definite integral produces a specific numerical value. In other words, the indefinite integral is used to find the antiderivative, while the definite integral is used to find the area under a curve.

4. Can an indefinite integral have multiple solutions?

Yes, an indefinite integral can have multiple solutions, as the antiderivative of a given function is not unique. This is because the derivative of a constant is always zero, so any constant value can be added to the antiderivative without changing its derivative.

5. What are the practical applications of evaluating an indefinite integral?

The evaluation of indefinite integrals has various practical applications in fields such as physics, engineering, economics, and statistics. It is used to calculate quantities such as displacement, velocity, acceleration, work, and cost, which are essential for understanding and analyzing real-world phenomena.

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