An Integral problem with x,lnx with progress done [ check it]

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In summary, the original formula had a denominator of 5 + u^2, but after solving the integral problem, the resulting expression used \sqrt{5} in the denominator instead. This change was made because the original formula diverged from the real formula. For more details and a complete solution, please refer to the scanned paper provided.
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Riazy
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An Integral problem with x,lnx with progress done [please check it]

QUESTION: How and what was changed from the original formula?
Whats up with all the Sqrts above the 5:s
Is this some kind of compensation because the current formula
diverge from the Real formula? How was it done?

Homework Statement


$dx/x(5*(lnx)^2), See a detailed scanned paper below


Homework Equations



You can find everything on the scanned paper below

The Attempt at a Solution



Yes The problem is solved already, but I have questions on why certain things are like they are. See the scanned paper. Thanks

Scanned Solution [PLAIN]http://img146.imageshack.us/img146/3544/kaat001.jpg
 
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The formula says that when the denominator is a2 + x2, the resulting expression uses a. The original expression has [itex]5 + u^2 = (\sqrt{5})^2 + u^2[/itex] in the denominator, so the resulting expression uses [itex]\sqrt{5}[/itex].
 

1. What is an integral problem with x, lnx?

An integral problem with x, lnx involves finding the function that, when differentiated, gives x, lnx. This is known as the antiderivative or indefinite integral of x, lnx. It is represented by ∫(x, lnx)dx and can also be written as ln(x^2/2) + C, where C is a constant of integration.

2. How do you solve an integral problem with x, lnx?

To solve an integral problem with x, lnx, you can use integration by parts or substitution. In integration by parts, you split the integral into two parts and use the formula ∫u dv = uv - ∫v du. In substitution, you replace the variable (in this case, x) with a new variable (usually u) and then solve for the new integral.

3. What is the purpose of solving an integral problem with x, lnx?

Solving an integral problem with x, lnx can help us find the area under a curve, which is useful in many real-world applications. It can also help us solve differential equations, which are fundamental in many fields of science and engineering.

4. Can you provide an example of an integral problem with x, lnx?

One example of an integral problem with x, lnx is ∫(x, lnx)dx = ∫xlnx dx. This can be solved using integration by parts, where u = ln x and dv = x dx. The solution is xlnx - ∫dx, which simplifies to xlnx - x + C.

5. How do you check if the progress in an integral problem with x, lnx is correct?

To check if the progress in an integral problem with x, lnx is correct, you can differentiate the solution and see if it gives x, lnx. In this case, the derivative of xlnx - x + C is ln x + 1, which is the same as x, lnx. Additionally, you can use online integration tools or graphing calculators to verify the solution.

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