CHAPTER Continuity Fundamental Theorem of Calculus In this lecture you will learn the most important relation between derivatives and areas (definite integrals).

Evaluating Theorem If f is continuous on the interval [a,b], then  a b f (x) dx = F (b) – F (a) where F is any antiderivative of f, that is, F’ = f.

Example Evaluate  0 1 ( y 9 + 2y 5 + 3y ) dy.

 f (x) dx = F (x) means F’ (x) = f (x) Indefinite Integrals: You should distinguish carefully between definite and indefinite integrals. A definite integral  b a f (x) dx is a number, whereas an indefinite integral  f (x) dx is a function.

Table of Indefinite Integrals  c f (x) dx = c  f (x) dx  [ f (x) + g(x)]dx =  f (x) dx +  g(x)dx  x n dx = (x n+1 ) / (n+1) + C (n  -1)  a x dx = (a x ) / (ln a) + C  e x dx = e x + C  (1/x) dx = ln |x| + C

Table of Indefinite Integrals  sin x dx = - cos x + C  cos x dx = sin x + C  sec 2 x dx = tan x + C  csc 2 x dx = - cot x + C  sec x tan x dx = sec x + C  csc x cot x dx = - csc x + C

Table of Indefinite Integrals  [ 1 / (x 2 + 1) ] dx = tan -1 x + C  [ 1 / (  1 - x 2 ) ] dx = sin -1 x + C ____

Example Find the general indefinite integral for  (cos x – 2 sin x) dx.

Example Find the general indefinite integral for:  [ x /(x 2 + 1)] dx.

Evaluating Theorem If f is continuous on the interval [a,b], then  a b f (x) dx = F (b) – F (a) where F is any antiderivative of f, that is, F’ = f.