Need some serious advice here or I'm in trouble(trust me)

[solitaire]

I have some problems with Intergration. Hope can get help in here. If I posted on wrong board, please forgive me.

My first problem is that in Integration by using Trigonometric identities, some function have more than one substitution. Like Sin(square).



SIN^2 xdx, SIN^2 can be substitude as:-


1 - COS2x/2,


and can also be substitude as :-




1 - COS^2 x



And in certain question, first substitution is used. While on others, the other substitution is use. I just want to know how can I recognise what substitution I've to use in which questions? Is there a simple to way to it?


And secondly. I don't know what's the problem with is, but in Examination Papers, the question regarding Integrations only say "Evaluate the Integral". They aren't specific about how, like they don't say "Evaluate the Integral by substitution method" or "Evaluate the Integral by Parts" or "Evaluate the Integral by Summation" etc. The way the just say "Evaluate" makes it very difficult for me to recognise which type of Integration method is required to solve the following problems, since most of them are just like each other. So if I can get some tips on how to recognise which type of Integration method is required for the certain equations, it will be quite a relief. I came here with some expectation, hope you guys can help me out alittle here.

 

[zhentil]

Oh man, you're going to hate this, but it's what I tell my students. Practice. Practice practice practice. This stuff is hard, and realizing that is the only way you're going to get better. If you don't know u-substitution and you're trying to do trig substitutions, you need to go back and practice until you understand.

On to your specific question, sin^2(x) should be evaluated using the cos(2x) substitution, because that has an elementary anti-derivative (sin(2x)/2). For all of these, the answer is: "Because it works!" If you can start viewing it like that, for instance, try to understand that the reason they introduce trig substitution is because u-substitution doesn't work for this new kind of problem. And integration-by-parts, trigonometric integrals, etc. are all introduced as means of solving problems you couldn't solve before.

Finally, the reason the exams don't tell you which method to use is because they're trying to test your ability to think and solve problems, not your ability to remember a formula. It's not a biology test.