Proof of Various Integral Facts

# Proof of Various Integral Facts - and we also have,...

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Proof of Various Integral Facts/Formulas/Properties In this section we’ve got the proof of several of the properties we saw in the Integrals Chapter as well as a couple from the Applications of Integrals Chapter. Proof of : where k is any number. This is a very simple proof. Suppose that is an anti-derivative of , i.e. . Then by the basic properties of derivatives we also have that, and so is an anti-derivative of , i.e. . In other words, Proof of : This is also a very simple proof Suppose that is an anti-derivative of and that is an anti-derivative of . So we have that and . Basic properties of derivatives also tell us that

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and so is an anti-derivative of and is an anti-derivative of . In other words, Proof of : From the definition of the definite integral we have,

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Unformatted text preview: and we also have, Therefore, Proof of : From the definition of the definite integral we have, Proof of : From the definition of the definite integral we have, Remember that we can pull constants out of summations and out of limits. Proof of : First well prove the formula for +. From the definition of the definite integral we have, To prove the formula for - we can either redo the above work with a minus sign instead of a plus sign or we can use the fact that we now know this is true with a plus and using the properties proved above as follows. Proof of : , c is any number. If we define then from the definition of the definite integral we have,...
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## Proof of Various Integral Facts - and we also have,...

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