a) integral of e^(x^2) (ie, integral of e to the x squared power)
b) integral from -1 to 6 of the square root of 3 plus the absolute power of x (sorry i don't know how to make it look nice on these forums...havent learned the code for it yet)
c) this one is slightly different: express the integral from 1 to 5 of x/(x+1) as a limit
The Attempt at a Solution
(a) integral of e^(x^2) (ie, integral of e to the x squared power)
i know the integral of e^x is itself. but i can't figure out how to adapt to the fact that the x is squared. i tried substitution but that didnt help much.
(b) integral from -1 to 6 of the square root of 3 plus the absolute power of x
since the function is always positive, i feel i can just take the integral of the square root of 3 + x (without the absolute power) and plug us -1 and 6 just like normal. is that correct?
(c) express the integral from 1 to 5 of x/(x+1) as a limit
always had a little trouble with these...
i know it will be: the limit as n -> oo (infinity) of the sum from i=1 to n of f(ci)*xi
(hope you can read that :) )
the f(ci) represents the heights of the infinite rectangles and the delta xi represents the equal widths of all the rectangles (and you just sum them all up).
but i always get confused on what to plug into the function itself. in this case x/(x+1). what is ci?
and also if xi = (b-a)/n (in this case 4/n) how does the integral change if it was from 2 to 6 because xi would again be 4/n? or does it not matter? i would think it should be different since the end points of the function change but xi seems to be the same.
thanks for any help!