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CHAPTER 2

CHAPTER 2. Fundamental Theorem of Calculus. 2.4 Continuity. 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

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CHAPTER 2

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  1. CHAPTER 2 Fundamental Theorem of Calculus 2.4 Continuity In this lecture you will learn the most important relation between derivatives and areas (definite integrals).

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

  3. Example Evaluate 01( y9 + 2y 5 + 3y) dy.

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

  5. Table of Indefinite Integrals • c f (x) dx = c f (x) dx  [ f (x) + g(x)]dx= f (x) dx + g(x)dx  xndx = (xn+1) / (n+1) + C (n  -1)  axdx = (ax) / (ln a) + C • exdx = ex + C (1/x) dx= ln |x| + C

  6. Table of Indefinite Integrals sin xdx = - cos x + C cos xdx = sin x + C sec2xdx = tan x + C  csc 2xdx = - cot x + C sec x tan x dx = sec x + C csc x cot x dx = - csc x + C

  7. Table of Indefinite Integrals [ 1 / (x2 + 1) ] dx = tan-1x + C [ 1 / ( 1 - x2 ) ] dx = sin-1x + C ____

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

  9. Example Find the general indefinite integral for:  [ x2+1 + 1/(x2 + 1)] dx .

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

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