Question

prove that the intgrale from a to b for
f(x)=x is =b^2 -a^2/2.

Answer 1

prove \[\int\limits_{a}^{b}x dx=b^{2}-a ^{2}/2\]

Answer 2

Proof
Strategy: Use the eu du = eu + C
since eln b = b,

bx dx = [ (eln b) x ] dx
= e (ln b) x dx

set u = (ln b) x
then du = (ln b) dx
substitute...

= eu (du / ln b)
= (1 / ln b) eu du

solve the integral...

= (1 / ln b) ( eu + C )
= (1 / ln b) eu + C2 (create new constant)

substitute back u = (ln b) x,

= ( 1 / ln b) e(ln b) x + C2
= ( 1 / ln b) ( e(ln b) )x + C2
= ( 1 / ln b) bx + C2
= bx / ln b + C2
Q.E.D.



2. You need not memorize this theorem. Derive it each time you use it.
Consider this example: if you have the integral:
2x dx.
There is no need to memorize the formula. We will get this integral into the easier form, eu du.
Recall that eln(2) = 2

2x dx = ( eln (2) ) x dx
= eln (2) x dx

set u = ln(2) x
then du = ln(2) dx
substitute:

= eu (du / ln 2 )
= (1 / ln 2) eu du
= (1 / ln 2) eu + C
substitute back...
= (1 / ln 2) eln(2) x + C
= (1 / ln 2) ( eln(2) )x + C
= (1 / ln 2) 2x + C ANSWER

Answer 3

deos that help

Answer 4

srry my key bored is bad

Answer 5

yes thanx ,but am still stuck how to prove it

Answer 6

umm i see i some time need help on that to umm

Answer 7

i kow hold on i know some won who may help

Answer 8

like my friend seid just follow the rule of integal :)

Answer 9

that is how

Answer 10

and he seid intgrale x dx = x^2/2

Answer 11

well that was better