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You are not logged in. #1 20060827 21:48:16
confused.....How do I integrate #2 20060828 04:11:42
Re: confused.....Have you learned the technique of Integration by Substitution? Here's the basic idea of it: In its current form, you probably don't recognize it as something you can integrate. but if it were in the form you could solve it in a heartbeat. The goal of Integration by Substitution is to get the integral into a simple form like the one above, so you can evaluate the integral. How do we go about this? Well, I'm not sure how well I can explain this, and it may seem like I am making so leaps in logic, but this is just a thing you need to practice, and then it will become natural and you'll be able to know the correct substitution to make nearly all of the time. Ok, now if we were to substitute some u in for x in a given integral ∫f(x) dx, by the chain rule the integral would turn into ∫f(u) du. This probably just looks confusing, doing the example will make it become clearer. So let's look at our integral here and find a suitable expression to set as u. The best way to go about this is to think of what integrals we can easily solve that look like the given integral and then what substitution x = u would be able to change the integral into that form. Looking at the integral, we may see the form which we know we can solve easily. It turns out that we can get this form if we let u = 5x²  1. We choose this substitution so that we can get only one term under the square root. Almost always with Integration by Substitution we wish to get only one term in places like (u)^{n}, cos(u), e^{u}, etc. Anyway, if we let u = 5x²  1, then du = 10x dx (do you get what I did here?). Now we plug in u for 5x²  1 and du for 10x dx(we multiplied 4x by 5/2 in the second step to get 4x to be 10x and to balance out that 5/2 we multiplied the entire integral by 2/5): Now we just substitute 5x²  1 for u to get our answer in terms of x: I hope this was clear. #3 20060828 04:47:15
Re: confused.....Thankyou, that was beautifully written. I actually learned alot from that. But what happened to the dx bit? It seemed to disappear. Shouldn't it have become 2/5 ∫ (u)^1/2 du dx? #4 20060828 04:59:48
Re: confused.....Remeber that du = 10x dx, so the dx is still in there, it's just hidden under the name of du. "In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..." #5 20060828 05:03:43
Re: confused.....Yes, Ricky is right. dx is part of du, so it "disappears" when we make the substitution du = 10x dx. #6 20060828 05:46:10
Re: confused.....You're a teacher? I should have known, there's difference between saying and teaching, it is a skill. Last edited by confused94 (20060828 05:48:28) #7 20060828 06:03:51
Re: confused.....Another question if I may, why do you multiply the whole integral by 2/5? #8 20060828 06:49:48
Re: confused.....Because to get 4x to be 10x, so we can substitute 10x dx = du, we need to multiply 4x by 5/2 ((5/2)*4x = 10x). But this gives us a different integral (instead of ∫f(x) dx we now have ∫(5/2)f(x) dx). So to keep the integral the same, we multiply it by the reciprocal of 5/2, which is 2/5. This will work out since (2/5)∫(5/2)f(x) dx = ∫f(x) dx, so it is indeed the same integral. Is that clear? Last edited by Zhylliolom (20060828 06:57:24) #9 20060828 06:54:06
Re: confused.....Also, note the following: since 4x/10x = 2/5. So from that, you have another way of seeing how the 2/5 showed up. #10 20060828 07:10:58
Re: confused.....Now I understand. In order for the integral to remain unchanged, you must multiply the by the reciprocal 2/5 * 5/2 = 1. Therefore it is unchanged. #11 20060828 07:41:00
Re: confused.....
Yea, sign me up as well. But on a serious note, the whole %10 thing is one big myth that was spread by the media. If you think you only use 10% of your brain, which part could you do without? "In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..." #12 20060828 12:04:05
Re: confused.....Here are some exercises(I recommend that you don't try the double starred ones until after Calculus III(I don't think most standard curriculums even teach the proper way to do the double starred ones, I just put them there for people like Ricky to pull their hair out over)): #13 20060830 07:18:46
Re: confused.....Hey confused94, just making sure you don't forget about the Integration by Substitution thread I made the other day in response to this post. Hopefully you can go to this thread and benefit from the practice on the problems. 