WEBVTT
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here we have a definite integral from zero to pie
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over two of co sign of tea divided by the
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square root of one plus sign square. Let's do
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it. Use up here and let's take you to
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be sci fi Then do you is cosign TT Since
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this is a definite no girl, we should switch
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these limits of integration So the new lower limit will
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become using this formula. Pierre. So we plug
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in t equals zero and we have u equals sign
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zero, which is zero and then for the upper
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limit. Piper too. So we have upper limit
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you equals signed by over two, which is one
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and then by Are you sub cosign t dt is
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just do you and then the denominator we have one
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plus you square inside the radical. So now for
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this new integral we should do entrance up. So
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let's take you equals Tan Daito Then do you a
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sequence where data and once again because we have Ah
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, definitely, girl. We should go out and
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switch these limits. So originally we had zero for
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you also Wait. One more observation since this is
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a definite no girl. Oh, when you're used
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when your tricks up is of the form eight and
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data the requirement is that date is between negative five
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for two and powerful too. So plugging in you
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equal zero to get a new lower limit, we
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have zero equals tan data. And if they does
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in this interval appear the only time tangent could be
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zero is a data equals zero. So that's our
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new lower limit for the upper limit. We have
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one equals tan data and the only time this happens
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in our honorable negative Piper tutto perverts Who is that
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power for some running on room here. So let
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me go to the next page. So we had
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that was the general. So now, after using
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our tricks up, we have sealed up IRA for
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do you with sequins. Where? Data. And
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in the bottom we have tan squared plus one.
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So using the Pythagorean identity for Tangent Square, we
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have seeking squared inside the radical and then the square
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root of seek and square as you see again so
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we could cross off one of the sequence Sierra Pi
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over four seek and data and we know this integral
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to be natural log, absolute value seek and data
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plus tan data. So this is a trick in
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a girl and there end points or zero power before
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. So if we plug in pi over four first
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, seeking a pi over four is rude, too
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. Tangent on power for is one and then seek
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in of zero is one. Tension of zero is
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zero and then natural log of one zero so we
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could ignore the second term and their final answer.
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Natural log one plus radical too. And you could
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lose the absolute value because one plus radical to his
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positive number.