Improper integral with substitution

In summary: So, what do you do with the sqrt(x)?After substituting u=√x, we have $$du = \frac 1 2 \frac{dx}{\sqrt{x}} \; \Rightarrow 2 du = \frac{dx}{\sqrt{x}}$$ and $$\frac{1}{x+1} =\frac{1}{u^2+1}.$$ Just put these together.In summary, the student is trying to find the improper integral by substituting one of the "elements" but doesn't understand how to get from one given step to the next. He is having difficulty understanding what to do
  • #1
ChristinaMaria
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Hi! I am trying to solve problems from previous exams to prepare for my own. In this problem I am supposed to find the improper integral by substituting one of the "elements", but I don't understand how to get from one given step to the next.

Homework Statement


Solve the integral
SrdRmDZ.png

by substituting u = sqrt(x)

Homework Equations


I don't understand how to get from step 1 to step 2:
HEnJgKc.png


The Attempt at a Solution


This is one of my attempts:
j6vAgRS.jpg

So, as I mentioned above I don't understand how to get from step one to two. I don't get what to do with the sqrt(x) you get in the expression dx = 2sqrt(x)du that I've written in the right corner. Where did it "go" in the step-by-step example? I can't seem to figure out how to remove it.

I hope this was easy enough to read.
Thanks :smile:
 

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  • #2
You seem to have turned x+1 into u+1 at the start of the second line.
 
Last edited:
  • Like
Likes ChristinaMaria
  • #3
ChristinaMaria said:
Hi! I am trying to solve problems from previous exams to prepare for my own. In this problem I am supposed to find the improper integral by substituting one of the "elements", but I don't understand how to get from one given step to the next.

Homework Statement


Solve the integral1
View attachment 235498
by substituting u = sqrt(x)

Homework Equations


I don't understand how to get from step 1 to step 2:
View attachment 235499

The Attempt at a Solution


This is one of my attempts:
View attachment 235500
So, as I mentioned above I don't understand how to get from step one to two. I don't get what to do with the sqrt(x) you get in the expression dx = 2sqrt(x)du that I've written in the right corner. Where did it "go" in the step-by-step example? I can't seem to figure out how to remove it.

I hope this was easy enough to read.
Thanks :smile:

From ##u = \sqrt{x}## we have
$$du = \frac 1 2 \frac{dx}{\sqrt{x}} \; \Rightarrow 2 du = \frac{dx}{\sqrt{x}}$$ and $$\frac{1}{x+1} =\frac{1}{u^2+1}.$$ Just put these together
 
  • Like
Likes ChristinaMaria
  • #4
ChristinaMaria said:
I don't get what to do with the sqrt(x) you get in the expression dx = 2sqrt(x)du that I've written in the right corner. Where did it "go" in the step-by-step example? I can't seem to figure out how to remove it.
When you change the variable, do not keep the old one.
You have u=√x, This means x=u2. What is dx/du? what is dx then?
 

FAQ: Improper integral with substitution

What is an improper integral with substitution?

An improper integral with substitution is an integral that cannot be evaluated using traditional methods, such as the fundamental theorem of calculus, because the limits of integration are infinite or the integrand is undefined at certain points. In these cases, a change of variables can be used to transform the integral into a form that can be evaluated.

When should substitution be used for improper integrals?

Substitution should be used for improper integrals when the integrand cannot be integrated using traditional methods, and the limits of integration are either infinite or the integrand is undefined at certain points. In these cases, substitution can simplify the integral and make it possible to evaluate.

How do you perform substitution for an improper integral?

To perform substitution for an improper integral, start by identifying the part of the integrand that is causing the improperness. Then, make a substitution by choosing a new variable that will cancel out the problematic part of the integrand. Finally, evaluate the integral using the new variable and the appropriate limits of integration.

What are some common substitutions used for improper integrals?

Some common substitutions used for improper integrals include trigonometric substitutions, exponential substitutions, and hyperbolic substitutions. The specific substitution used will depend on the form of the integrand and the limits of integration.

Can substitution always be used for improper integrals?

No, substitution cannot always be used for improper integrals. There are some cases where substitution will not work, such as when the integrand is undefined at all points or when the limits of integration are not finite. In these cases, other methods, such as integration by parts or partial fractions, may need to be used.

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