Integrating x√(x^2+a^2) using Substitution Method

In summary, the conversation discusses evaluating the integral of x * sqrt(x^2 + a^2) with limits from 0 to a. The individual suggests using substitution with u = x^2 + a^2 and dropping the a^2 term, but is unsure if this approach is correct. They are reassured that it is correct as long as a is a constant and reminded to consider the corresponding u-limits. The conversation ends with the individual thanking for the help and gaining more confidence in solving similar problems.
  • #1
sapiental
118
0
Hello,

evaluate the following integral:

[tex]\int x \sqrt{x^2+a^2}dx [/tex]

definite integral from 0 to a

what I did was

u = x^2 + a^2
du = 2xdx

1/2 sqrt(u)du

I just dropped the a^2 because we were finding the derivative of x but feel that it's very wrong.Any suggestions are much appreciated.

thanks.
 
Last edited:
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  • #2
sapiental said:
...
I just dropped the a^2 because we were finding the derivative of x but feel that it's very wrong.

It's correct, if a is a constant.
 
  • #3
Why do you feel it is wrong?
Ask yourself:
1. What is the derivative of a constant?

2. What are the corresponding u-limits compared to the given x-limits?
 
  • #4
You'll notice that when you go from u to x again, the a2 term pops back in.

So it's not like you completely lost it
 
  • #5
hey,

thanks again for all the help. I learn so much from this website and it helps me be much more confident with the problems.

d/dx of C = 0

and the corresponing u limits are 0 + a^2 and a^2 + a^2

what throws me off about dropping the a^2 is that a itself is one of the limits of integration.

thanks!
 
  • #6
Why should that matter??
Would you have accepted it if the limit was some number c instead?
 
  • #7
ah ok, I understand now. thanks again.
 

1. What is integration by substitution?

Integration by substitution, also known as u-substitution, is a technique used in calculus to simplify the process of integration. It involves replacing a complex expression with a new variable, u, and then integrating with respect to u instead of the original variable.

2. When should I use integration by substitution?

Integration by substitution is most commonly used when dealing with integrals that involve a product of two functions or a function raised to a power. It is also helpful when the integral cannot be solved by other techniques, such as integration by parts.

3. How do I choose the substitution variable?

The substitution variable, u, should be chosen in a way that simplifies the integral. This can be done by looking for a function within the integral that resembles the derivative of another function. It is also helpful to choose a substitution that cancels out any complicated terms or fractions.

4. What are some common mistakes to avoid when using integration by substitution?

Some common mistakes to avoid include forgetting to substitute back in the original variable, not properly calculating the derivative of the substituted function, and not changing the limits of integration. It is also important to be careful with algebraic manipulations and to always check your answer.

5. Are there any limitations to integration by substitution?

Integration by substitution may not always work for all integrals. In some cases, the substitution may make the integral more complicated or impossible to solve. It is important to practice and gain experience in recognizing when this technique will be helpful.

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