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In T Integral Sin – How To Integrate Sin^2 X

Before proceeding with the proof, we recall the second mean-value theorem for integrals (Theorem 5.5 on p. 219 of Apostol). For a continuous function

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on the interval ” title=”Rendered by QuickLaTeX.com” height=”16″ width=”29″ style=”vertical-align: -4px;”/> if

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has a continuous derivative which never changes sign on the interval ” title=”Rendered by QuickLaTeX.com” height=”16″ width=”29″ style=”vertical-align: -4px;”/> then there exists a

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” title=”Rendered by QuickLaTeX.com” height=”16″ width=”55″ style=”vertical-align: -4px;”/> such that

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” title=”Rendered by QuickLaTeX.com”/>

Proof. Now, we want to apply the mean-value theorem above with

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and

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. Since

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is continuous everywhere

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, it is continuous on any interval

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” title=”Rendered by QuickLaTeX.com” height=”16″ width=”30″ style=”vertical-align: -4px;”/>. Then,

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” title=”Rendered by QuickLaTeX.com”/>

is continuous for all

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(so, in particular, for all ). Furthermore, since

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0″ title=”Rendered by QuickLaTeX.com” height=”18″ width=”79″ style=”vertical-align: -4px;”/> for all we have that

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But since

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for all we know

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for any

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and . Hence,

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” title=”Rendered by QuickLaTeX.com”/>

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5 comments

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June 20, 2017 at 12:16 pm
Andres Tellez says:

At the end, following the same procedure, you get (x(1-cosc)+1-cosc)/(x+1) which is always greater or equal to 0 for any choice of c or x, such that 0Reply

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May 4, 2016 at 6:42 pm

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If you are having trouble with math proofs a great book to learn from is How to Prove It by Daniel Velleman:

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