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Let ''ƒ'' be a continuous real-valued function defined on a closed interval [''a'', ''b'']. Let ''F'' be the function defined, for all ''x'' in [''a'', ''b''], by | Let ''ƒ'' be a continuous real-valued function defined on a closed interval [''a'', ''b'']. Let ''F'' be the function defined, for all ''x'' in [''a'', ''b''], by | ||
: | :[[File:FTC1.png]] | ||
Then, ''F'' is continuous on [''a'', ''b''], differentiable on the open interval (''a'', ''b''), and | Then, ''F'' is continuous on [''a'', ''b''], differentiable on the open interval (''a'', ''b''), and | ||
: | :[[File:FTC2.png]] | ||
for all ''x'' in (''a'', ''b''). | for all ''x'' in (''a'', ''b''). | ||
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The fundamental theorem is often employed to compute the definite integral of a function ''ƒ'' for which an antiderivative ''g'' is known. Specifically, if ''ƒ'' is a real-valued continuous function on [''a'', ''b''], and ''g'' is an antiderivative of ''ƒ'' in [''a'', ''b''], then | The fundamental theorem is often employed to compute the definite integral of a function ''ƒ'' for which an antiderivative ''g'' is known. Specifically, if ''ƒ'' is a real-valued continuous function on [''a'', ''b''], and ''g'' is an antiderivative of ''ƒ'' in [''a'', ''b''], then | ||
: | :[[File:FTC3.png]] | ||
The corollary assumes continuity on the whole interval. This result is strengthened slightly in the following theorem. | The corollary assumes continuity on the whole interval. This result is strengthened slightly in the following theorem. | ||
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Let ''ƒ'' be a real-valued function defined on a closed interval [''a'', ''b''] that admits an antiderivative ''g'' on [''a'', ''b'']. That is, ''ƒ'' and ''g'' are functions such that for all ''x'' in [''a'', ''b''], | Let ''ƒ'' be a real-valued function defined on a closed interval [''a'', ''b''] that admits an antiderivative ''g'' on [''a'', ''b'']. That is, ''ƒ'' and ''g'' are functions such that for all ''x'' in [''a'', ''b''], | ||
: | :[[File:FTC4.png]] | ||
If ''ƒ'' is integrable on [''a'', ''b''] then | If ''ƒ'' is integrable on [''a'', ''b''] then | ||
: | :[[File:FTC5.png]] | ||
Notice that the Second part is somewhat stronger than the Corollary because it does not assume that ''ƒ'' is continuous. | Notice that the Second part is somewhat stronger than the Corollary because it does not assume that ''ƒ'' is continuous. |