Evaluate ∫ (3 + √x) / (x(3 + x)) dx

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In summary, the process for evaluating this integral involves using the substitution method, specifically substituting u = √x and using partial fractions. The domain of this integral is all real numbers except for x = 0 and x = -3. It can also be solved using integration by parts, but the substitution method is typically more efficient. There is no limit to the number of substitutions that can be used, but it is important to choose ones that simplify the integral. The square root in the numerator indicates that the substitution method is necessary to solve the integral.
  • #1
devonsn
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Evaluate the integral ∫ (3 + √x) / (x(3 + x)) dx

I've broken it up into ∫ [3 / x(3 + x)] dx + ∫ [ √x / (x(3 + x))] dx

And that's as far as I can get.
 
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  • #2
The first one is partial fractions. For the second one I suggest u=sqrt(x).
 

1. What is the process for evaluating this integral?

The process for evaluating this integral involves using the substitution method. First, substitute u = √x, which will transform the integral into ∫ (3 + u) / (u^2 + 3) du. Then, use partial fractions to break the expression into easier integrals. Finally, solve for the values of the integrals and substitute back in the original variable x to get the final answer.

2. What is the domain of this integral?

The domain of this integral is all real numbers except for x = 0 and x = -3. This is because the denominator cannot be equal to 0 in order for the integral to be defined.

3. Can this integral be solved using any other method?

Yes, this integral can also be solved using integration by parts. However, the substitution method is typically easier and more efficient for this specific integral.

4. Is there a limit to the number of substitutions that can be used in this integral?

No, there is no limit to the number of substitutions that can be used. However, it is important to choose substitutions that will simplify the integral and make it easier to solve.

5. What is the significance of the square root in the numerator of this integral?

The square root in the numerator indicates that the substitution method will be necessary to solve the integral. This is because the square root cannot be integrated directly and must be transformed into a form that can be integrated.

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