Integration of irrational function

In summary, the conversation is about a question involving the integral of sqrt[x/(a-x)]. The person tried to use a substitution of u=sqrt[x/(a-x)] and dx= 2au/(1+u2)2 du. After some steps, they obtained an answer with tan-1, but the model answer given did not have tan-1. They asked for clarification on whether their approach was correct and if the difference in the final answer was due to a constant of integration or a wrong substitution. They were advised to check their work on the substitution of u=sqrt[x/(a-x)].
  • #1
KLscilevothma
322
0
Here's the question that I got stuck:

[inte]sqrt[x/(a-x)] dx .........(*)

I tried to use the following substitution
u=sqrt[x/(a-x)] and ........(1)
dx = 2u(1-a)/(1+u2)2 du...(2)

sub (1) and (2) into (*), after a few steps, I got

(2-2a)[inte]du/(1+u2) - 2(1-a)[inte]du/(u2+1)2

The answer derived from the first part, (2-2a)[inte]du/(1+u2), contains tan -1 but the model answer of this question is
-[squ](ax-x2) + a/2sin-1[(2x+a)/a] + C
For the second part, I let u = tan θ and got a strange expression.

Is my approach correct and is the final answer obtained from the above method differs the model answer only by the constant of integration ? Or am I using a wrong substitution?
 
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  • #2
Check your work on your substitution.
 
  • #3
u=sqrt[x/(a-x)]

dx = 2au/(1+u2)2 du

thanks
 

1. What is an irrational function?

An irrational function is a mathematical function that contains irrational numbers, meaning numbers that cannot be expressed as a ratio of two integers. Examples of irrational numbers include pi and the square root of 2. Irrational functions often involve radicals, logarithms, and trigonometric functions.

2. How is integration of irrational functions different from integration of rational functions?

Integrating irrational functions can be more challenging than integrating rational functions because irrational functions do not have a finite number of terms. This means that the integration process can be more complex and require advanced techniques such as substitution or integration by parts.

3. What are some common strategies for integrating irrational functions?

Some common strategies for integrating irrational functions include using substitution, integration by parts, and trigonometric identities. It is also important to simplify the function as much as possible and to use known integration techniques for specific types of irrational functions, such as logarithmic or trigonometric functions.

4. Can irrational functions be integrated using the Fundamental Theorem of Calculus?

Yes, the Fundamental Theorem of Calculus can be used to integrate irrational functions. However, it may require some additional steps and techniques, as mentioned in the previous questions. It is a powerful tool for finding the antiderivative of a function and can be applied to both rational and irrational functions.

5. How important is it to check the result of an integrated irrational function?

It is essential to check the result of an integrated irrational function, as it is easy to make mistakes and overlook certain steps in the integration process. Checking the result can also help identify any algebraic errors or incorrect substitutions. It is also a good practice to check the result by differentiating the integrated function and comparing it to the original function.

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