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Saterial
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Homework Statement
∫√(1+x2)/x dx
Homework Equations
The Attempt at a Solution
Let u = 1+x2
du = 2xdx
1/2du=xdx
x=√(u-1)
∫√1+x2/x dx
=∫√u/√(u-1) du
or is it 1/2∫√udu as xdx would remove it. This is where I got confused.
What did you do with the x in the denominator ofSaterial said:Homework Statement
∫√(1+x2)/x dx
Homework Equations
The Attempt at a Solution
Let u = 1+x2
du = 2xdx
1/2du=xdx
x=√(u-1)
∫√1+x2/x dx
=∫√u/√(u-1) du
or is it 1/2∫√udu as xdx would remove it. This is where I got confused.
You're missing an x.Saterial said:In my solution attempt, I tried two different ways. In your specific question, I replaced the x in the denominator with √(u-1) , as solving for x in u=1+x^2
There are two ways that I know of to do this integration.Saterial said:Thanks for the help so far. With that little fix, I've now reached the point of
1/2 ∫√u/(u-1) du
I don't know where to go from here. I also am having many people tell me that this method will not work? That Trig-Substitution is the only way to solve this problem... Is that true? Am I just wasting my time?
The purpose of integrating √(1+x2)/x is to find the area under the curve of the function. Integration is a mathematical process that allows us to find the total value of a function over a given interval.
A step-by-step guide is necessary because integration can be a complex process and involves multiple steps. It is important to follow a systematic approach to ensure the correct solution is obtained.
The steps involved in integrating √(1+x2)/x are:
1. Identify the integral and rewrite it in the correct form
2. Use a substitution to simplify the integral
3. Apply the substitution rule to find the new integral
4. Solve the new integral using techniques like integration by parts or trigonometric substitution
5. Simplify the final solution and include the constant of integration.
Yes, there are special cases to consider when integrating √(1+x2)/x. One such case is when the integral is improper, meaning it has an infinite limit or an undefined value. In these cases, additional steps may be required to solve the integral.
You can check your solution by differentiating the answer and comparing it to the original function. If they are the same, then your solution is correct. Additionally, you can use a graphing calculator to plot the original function and your solution to visually confirm if they are the same.