How Do You Integrate (x^(1/2))/ln(x) Using Integration by Parts?

In summary, the question is asking for the improper integral of (x^1/2)/lnx dx. The individual attempted methods were trying to solve it by parts and substituting x=e^2t, but neither were successful. The expert suggests applying parts directly in a specific form to involve the derivative of (1/lnx), which should lead to a simpler evaluation of the integral.
  • #1
Engineerbrah
5
0
QUESTION:

The question is to find the improper integral of (x^1/2)/lnx dx.

MY ATTEMPT:
1)I tried it byparts, by taking 1/ln x as 'u' or the first function but i got stuck.

2)Alternatively, I tried substituting x=e^2t in hopes to eliminate ln for a simpler byparts integration, but that didn't work
out.

Please help asap. Thanks.
 
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  • #2
Engineerbrah said:
QUESTION:

The question is to find the improper integral of (x^1/2)/lnx dx.

MY ATTEMPT:
1)I tried it byparts, by taking 1/ln x as 'u' or the first function but i got stuck.

2)Alternatively, I tried substituting x=e^2t in hopes to eliminate ln for a simpler byparts integration, but that didn't work
out.

Please help asap. Thanks.
You can apply parts directly. Apply parts in the form such that the evaluation of the new integral involves the derivative of (1/lnx).
 
  • #3
Devin said:
You can apply parts directly. Apply parts in the form such that the evaluation of the new integral involves the derivative of (1/lnx).

I tried it this way. The furthest I got was

(2*(x)^(1/2))/lnx - Integral of (2/((x)^(1/2))(lnx)^2 dx

Still not able to attain the answer.
 
  • #4
Engineerbrah said:
I tried it this way. The furthest I got was

(2*(x)^(1/2))/lnx - Integral of (2/((x)^(1/2))(lnx)^2 dx

Still not able to attain the answer.
Devin said:
You can apply parts directly. Apply parts in the form such that the evaluation of the new integral involves the derivative of (1/lnx).
I believe this to be the way to approach. Then the evaluation of the new integral is straightforward.

<Mod note: attachment deleted>
 
Last edited by a moderator:

1. How do I know when to use integration by parts?

Integration by parts is used when the integral involves the product of two functions. In general, the formula for integration by parts is ∫uv' dx = uv - ∫u'v dx. So, if the integral you are trying to solve has a product of two functions, then you should consider using integration by parts.

2. What is the process for solving an integral by parts?

The process for solving an integral by parts involves choosing a function to differentiate, and another function to integrate. This is usually done using the acronym LIATE, which stands for Logarithmic, Inverse trigonometric, Algebraic, Trigonometric, and Exponential. The function you choose to differentiate is typically the first function in LIATE, and the function you choose to integrate is the second function in LIATE. Once you have chosen the functions, you can apply the integration by parts formula and continue to integrate until you reach a final answer.

3. What should I do if I get stuck while solving an integral by parts?

If you get stuck while solving an integral by parts, you can try using the tabular method. This method involves creating a table with two columns, one for the functions and one for their derivatives. Then, you can use the table to determine a pattern and continue integrating until you reach a final answer.

4. Can I use integration by parts for definite integrals?

Yes, integration by parts can be used for definite integrals as well as indefinite integrals. When solving a definite integral, you will need to evaluate the integral using the fundamental theorem of calculus after applying the integration by parts formula.

5. Are there any common mistakes to avoid when using integration by parts?

One common mistake to avoid when using integration by parts is not choosing the correct functions to differentiate and integrate. It is important to follow the LIATE rule and choose the functions in the correct order. Another mistake to avoid is forgetting to include the minus sign in the integration by parts formula. It is also important to be careful with algebraic manipulations when integrating multiple times.

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