- #1
lo2
I find this one tough so could anyone please help me solving this intregral:
[tex]\int{\sqrt{x^2-1}}dx[/tex]
I would like the procedure.
[tex]\int{\sqrt{x^2-1}}dx[/tex]
I would like the procedure.
Robokapp said:I've seen this before. Ok first keep in mind that you are integrating a semicircle...so...you can cheat and just geometrically find the area...or you can use a u-substitution.
Robokapp said:In "conic Sections" you learn this as being a hyperbola. It would be a circle if it had a +. But you only deal with half of it so circle or hyperbola is same thing...the answer would be in terms of Pi but integrals are not exact either...
lo2 said:I find this one tough so could anyone please help me solving this intregral:
[tex]\int{\sqrt{x^2-1}}dx[/tex]
I would like the procedure.
tieu said:ummm has any 1 solved it. if not isn't it just
1/3 * 1/(2x) (x^2 - 1)^(3/2)
= 1/(6x)(x^2 - 1)^(3/2)
correct me if I am wrong :)
VietDao29 said:There's another way though...
Ok, can you get it? :)
Yes, I agree to your opinion. Using cosh, or sinh substitution is much faster. However, what if the OP hasn't learned hyperbolic functions?uart said:Well if you look at what you've done it really was pretty much the same as what I did. Only the first step in your solution {to get the integral of 1/sqrt(x^2+a) } was different. But personally I prefer to use a sinh or cosh substitution (as appropriate depending on the sign of "a") to do that one. I think it's easier.
Robokapp said:Now the way you form a circle...or a hyperbola is by having 2 parabolas. Correct? Well here we have one...so it can be a part of a circle or of a hyperbola. It doesn't matter because the other part is not something we are looking at.
So by considering half of a hyperbola...wouldn't we consider a parabola?
And does that parabola not look like ... a parabola that would form a circle if theere would be its negative counterpart?
All I'm trying to do (i know i go way too far and I'm sorry for that) is try to set it up in terms of Pi. I don't know anything about hyperbolas. I don't even know if they have areas. I don't know if parabolas do either...but i know circles and semicircles do. I'm just relating it to something I can work with.
Robokapp said:I am not writing gibberish on purpose...when I write my paragraphs I am sure that what I say is at least worth reading. I'm being told it's not, and I'm trying to fix that, by asking questions which create more confusion.
Now how is a parabola a half circle part:
-Edited-
I worked the math...I was saying a big stupidity. I feel awful now... I was not doing an operation to both sides of the equal as I should have. I was getting one power less on y.
Could have been mine. I had posted (or thought I did anyway), but I guess it was deleted as being off-topic.d_leet said:I could have sworn there was another post here this morning in response to mine,
WhyIsItSo said:Could have been mine. I had posted (or thought I did anyway), but I guess it was deleted as being off-topic.
A tough integral problem is a mathematical equation that requires advanced techniques to solve. It involves finding the area under a curve, where the curve is a complicated function that cannot be easily integrated using basic methods.
Tough integral problems are difficult to solve because they involve complex functions with no simple algebraic solutions. This means that standard integration techniques such as substitution or integration by parts may not work, requiring more advanced methods.
Some strategies for solving tough integral problems include using special integration techniques such as partial fractions, trigonometric substitutions, or integration by parts. Another approach is to use numerical integration methods, such as Simpson's rule, to approximate the solution.
To improve your skills in solving tough integral problems, it is important to have a strong foundation in basic integration techniques and mathematical concepts such as trigonometry and calculus. Additionally, practicing with a variety of problems and seeking help from resources such as textbooks, online tutorials, and peers can also be helpful.
Yes, there are many real-world applications for solving tough integral problems. For example, in physics and engineering, integration is used to calculate areas, volumes, and other physical quantities. In economics and finance, integration is used to determine the total profit or cost of a business. Additionally, integration is also used in statistics and probability to find the probability of events occurring within a given range.