I tried to evaluate the integral , where by making the usage of the Reimann sum, but i failed.
More precisely, how should I pick and below?
What level maths are you taking?
Anyway, incase you were wondering, you can prove Integration by Parts using the Product Rule.
If you can prove this relationship works, I'm sure you'd be allowed to use it.
If and , by the product rule...
I think it's enough to provide a proof, using Riemann sums, that
and then to directly evaluate .
Technically speaking, if you're evaluating a definite integral, you ARE using a Riemann sum.
I think you're making the problem harder than it needs to be...
OK, I've done it, but I don't think you'll like the solution...
For this to work, we need to make use of the Mean Value Theorem.
i.e. for some function , which is continuous on , we have
, where and .
We can rearrange the formula so that it is of the form
For this function, , suppose we take some rectangular strip, between and .
Then this strip's area is
, for some .
Using the Mean Value Theorem, we can rewrite this as
, where .
Clearly, since , this means .
So the area of one strip is...
If we have strips, then the Area between and is
, since this is a telescopic series.
Since and , you have
To fully complete this, you would need to note that you make and then take the limit.
Can you evaluate
Something similar was posed a good while back. Here is the link if you're interested.
I think you did not read what I wrote in the topic, I dont want to calculate the integral with formulas, I want to show that the Reimann sum converges to .
If you dont have any answer, please dont mess the topic, but if you really want to help then you are welcome.