Improper integral with substitution

[ChristinaMaria]

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

by substituting u = sqrt(x)

Homework Equations

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

The Attempt at a Solution

This is one of my attempts:

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

 

[haruspex]

You seem to have turned x+1 into u+1 at the start of the second line.

 

[Ray Vickson]

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

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

 

[ehild]

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?