Change of variables in an integral

In summary, the conversation discusses the process of changing variables in integrals and calculating the limits of the new integral. The example of ##\int_{-1}^{0} x(x^2-1) dx## is used to explain how the lower limit is determined by plugging in the value of the original variable in the new variable. The conversation also mentions using the fact that ##\int_a^b f(x)dx= -\int_b^a f(x)dx## to adjust for the lower limit being larger than the upper limit. The topic of changing variables is further explored with the example of ##\int_{-1}^{0} x dx## and the use of the variable ##u^2 =
  • #1
2sin54
109
1
This is not really a homework or anything, just found myself hitting a wall when doodling around.
If I have an integral like

##\int_{-1}^{0} x(x^2-1) dx##

and I introduce a new variable:
##u = x^2##
How do I calculate the limits of the new integral? In this case the upper limit is obviously 0, but what about the lower limit?
 
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  • #2
For the lower limit you plug in -1 in that u=x^2 and so you have u=1... same thing for the upper limit. Of course you don't necessarily have to change the variable for this integral...
 
  • #3
When x= -1, u= (-1)2= 1, of course. If it bothers you that the lower limit of integration is larger than the upper limit just use the fact that [itex]\int_a^b f(x)dx= -\int_b^a f(x)dx[/itex].
 
  • #4
Argh, that was a brain fart.. But what about a case where
##\int_{-1}^{0} x dx##
and I make a change of variables like:
##u^2 = x##
Does the lower limit become a complex number?
 
  • #5
Yes, and, in fact, you would have to consider the case of u= i and u= -i separately. Further you would have to consider that in the complex numbers there are an infinite number of paths between i and 0 or between -i and 0! Not every substitution is a good idea.
 

FAQ: Change of variables in an integral

What is a change of variables in an integral?

A change of variables in an integral is a method used in calculus to simplify the evaluation of integrals. It involves substituting a new variable for the original variable in the integral, which can often make the integral easier to solve.

Why is a change of variables useful in solving integrals?

A change of variables can make it easier to solve integrals by transforming the integral into a more manageable form. It can also help to reveal patterns and relationships between different integrals.

What are the steps to perform a change of variables in an integral?

The steps to perform a change of variables in an integral are: 1) Choose a new variable to substitute for the original variable in the integral, 2) Determine the relationship between the new variable and the old variable, 3) Substitute the new variable and its corresponding differential into the integral, 4) Simplify the integral using the new variable, and 5) Transform the limits of integration to match the new variable.

What are some common substitutions used in a change of variables?

Some common substitutions used in a change of variables include trigonometric functions, exponential and logarithmic functions, and hyperbolic functions. Other substitutions may also be used depending on the specific integral being solved.

When should a change of variables be used in solving an integral?

A change of variables should be used when the original integral is difficult to solve or does not have a closed form solution. It can also be used to simplify integrals with complicated expressions or to reveal hidden relationships between integrals.

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