Simple Integration using U Substitution

In summary, the student attempted to solve an equation using a substitution and ended up with the correct answer.
  • #1
anon413
13
0

Homework Statement


Find the indefinite integral.
The antiderivative or the integral of (x^2-1)/(x^2-1)^(1/2)dx



Homework Equations





The Attempt at a Solution



Tried using (x^2-1)^(1/2) as u and udu for dx and I solved for x but I am still left with a 1 on top not sure how to deal with it.


 
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  • #2
[tex]\int\frac{x^2-1}{\sqrt{2x-1}}dx[/tex]

Yes?
 
Last edited:
  • #3
Sorry made a mistake in typing the denominator (2x-1)^(1/2) I am slightly displexic
 
  • #4
How did you type that exactly.
 
  • #6
Yes thank you, that command latex is complicated I'll try to learn it.
 
  • #7
well try to use this substitution:
2x-1=u^2, try to defferentiate and substitute back what u get for dx,
you also get for x=(u^2 + 1)/2, from 2x-1=u^2
the rest is pretty simple after u substitute!
Can you go from here?
 
  • #8
anon413 said:
Yes thank you, that command latex is complicated I'll try to learn it.
Gets easier as you use it on a daily basis.

First, I would break up the numerator.

[tex]\int\frac{x^2}{\sqrt{2x-1}}dx-\int\frac{dx}{\sqrt{2x-1}}[/tex]
 
  • #9
rocophysics said:
Gets easier as you use it on a daily basis.

First, I would break up the numerator.

[tex]\int\frac{x^2}{\sqrt{2x-1}}dx-\int\frac{dx}{\sqrt{2x-1}}[/tex]

It might work this way also, but by immediately taking the substitution that i suggested he will get to the result pretty fast.
 
  • #10
Ok ill try stupid maths method I would get the integral of (((u^2+1)/2)^2-1)/(u) what would my du be
 
Last edited:
  • #11
I guess I will attempt this on my own that's for the help.
 
  • #12
You do not need to know what the du will be, you need just to plug in the value of the dx that you get after differentiating 2x-1=u^2, so it obviously will be dx=udu, and when you plug this in the u here and that one that you will get on the denominator will cancel out so you are left with somehting like this:
[tex]\int\frac{((\frac{u^{2}+1}{2})^{2}-1)udu}{u}[/tex], now you can go from here right?
 
  • #13
anon413 said:
Ok ill try stupid maths method I would get the integral of (((u^2+1)/2)^2-1)/(u) what would my du be
By the way it is not stupid math, but instead sutupidmath! NOt that i mind it, but it feels good to be correct! NO hard feelings, ok?
 
  • #14
thx stupidmath. I knew I got the right udu for dx.
 

1. What is U Substitution?

U Substitution is a technique used in integration to simplify and solve integrals. It involves substituting a new variable, typically denoted as u, in place of a complex expression within the integral.

2. When should I use U Substitution?

U Substitution is most commonly used when the integrand (the expression being integrated) is composed of two functions, one of which is nested inside the other. It can also be used when the integrand contains a polynomial, trigonometric, or exponential function.

3. How do I perform U Substitution?

The first step in performing U Substitution is to identify the nested function within the integrand and assign it as the new variable u. Then, you must find the derivative of u and substitute it into the integrand, along with du, the differential of u. From there, the integral can be solved using standard integration techniques.

4. What are the benefits of using U Substitution?

U Substitution can simplify complex integrals, making them easier to solve. It can also help to identify patterns and relationships between different types of integrals, which can be useful in solving more difficult integration problems.

5. Are there any limitations to using U Substitution?

U Substitution may not always be applicable, as it requires the integrand to contain a nested function. It also may not work for integrals with multiple variables or those that cannot be expressed in terms of u. In these cases, other integration techniques, such as integration by parts, may be more suitable.

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