Chapter 5 | Integration
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Second, it is worth commenting on some of the key implications of this theorem. There is a reason it is called the Fundamental Theorem of Calculus. Not only does it establish a relationship between integration and differentiation, but also it guarantees that any integrable function has an antiderivative. Specifically, it guarantees that any continuous function has an antiderivative. Proof Applying the definition of the derivative, we have
F ( x + h )− F ( x ) h
F ′( x ) = lim h →0
⎡ ⎣ ⎢ ∫ ⎡ ⎣ ⎢ ∫
⎤ ⎦ ⎥
x + h
x
1 h 1 h
f ( t ) dt − ∫ f ( t ) dt + ∫
f ( t ) dt
= lim
h →0
a
a
⎤ ⎦ ⎥
x + h
a
f ( t ) dt
= lim
h →0
a
x
x + h
1 h ∫ x
f ( t ) dt .
= lim
h →0
Looking carefully at this last expression, we see 1 h ∫ x x + h
f ( t ) dt is just the average value of the function f ( x ) over the
interval ⎡ ⎣ x , x + h ⎤ ⎦ . Therefore, by The Mean Value Theorem for Integrals , there is some number c in ⎡ ⎣ x , x + h ⎤ ⎦ such that 1 h ∫ x x + h f ( x ) dx = f ( c ). In addition, since c is between x and x + h , c approaches x as h approaches zero. Also, since f ( x ) is continuous, we have lim h →0 f ( c ) = lim c → x f ( c ) = f ( x ). Putting all these pieces together, we have
x + h
1 h ∫ x
F ′( x ) = lim h →0
f ( x ) dx
f ( c )
= lim = f ( x ), h →0
and the proof is complete. □ Example 5.17 Finding a Derivative with the Fundamental Theorem of Calculus Use the Fundamental Theorem of Calculus, Part 1 to find the derivative of g ( x ) = ⌠ ⌡ 1 x 1 t 3 +1 dt . Solution According to the Fundamental Theorem of Calculus, the derivative is given by g ′( x ) = 1 x 3 +1 .
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