Struggling with an Integral? Try These Substitution Strategies!

  • Thread starter adelin
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In summary, the conversation is about finding a substitution for the integral ∫dx/2√x+2x. The suggestion is to convert the addition form in the denominator into a product form to make the function and its derivative more clear. The goal is to get rid of the square root.
  • #1
adelin
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I cannot find a substitution that work for this integral

∫dx/2√x+2x

what should I do?
 
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  • #2
What is [itex] \left(\sqrt{x}\right)^2[/itex]?
 
  • #3
it is∫dx/2sqrt(x)+2x
 
  • #4
See what you can do to bring a function and it's derivative.. you have √x in the denominator so you should think in terms of derivative of that.
 
  • #5
If the integral is
[tex]
\int \frac{1}{2 \sqrt{x} + 2x} \, dx
[/tex]

thinking about the derivative of [itex] \sqrt{x} [/itex] alone will not help.
 
  • #6
yes, this is the integral
 
  • #7
If he tries to bring the derivative of √x so that it is in product with the original function √x, he can substitute easily. That's what I meant.
 
  • #8
Convert the addition form in the denominator into product form so that the function and it's derivative are seen clearly in multiplied form.
 
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  • #9
Then think about the obvious substitution, aiming to get rid of the square root :-).
 
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  • #10
thank you
 

1. How do I know which method to use to solve an integral?

There are several methods for solving integrals, including substitution, integration by parts, and partial fractions. The best way to determine which method to use is to look at the integrand and try to identify any patterns or similarities to known integrals. Practice and experience will also help in choosing the most efficient method.

2. Can I use a calculator to solve an integral?

While calculators can be helpful for checking your work, it is important to learn how to solve integrals by hand. Many integrals require knowledge of algebraic and trigonometric identities, which calculators may not be able to handle. Additionally, understanding the steps to solving an integral will deepen your understanding of the concept.

3. What is the importance of the constant of integration?

The constant of integration represents all possible solutions to an indefinite integral. It is added to the end of the integral to account for any possible values that could make the integral true. Without the constant of integration, the solution to an indefinite integral would not be complete.

4. Can I solve an integral without using any known methods?

In some cases, integrals cannot be solved using traditional methods. In these cases, numerical methods such as Simpson's rule or the trapezoidal rule can be used to approximate the value of the integral. However, these methods are not always accurate and should only be used as a last resort.

5. How can I check if my solution to an integral is correct?

Once you have solved an integral, you can check your work by taking the derivative of your solution. If the derivative matches the original integrand, then your solution is correct. You can also use online integral calculators or ask a fellow mathematician to verify your solution.

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