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Problem 29 Easy Difficulty

Evaluating a Definite Integral In Exercises $21-32$ evaluate the definite integral.
$$
\int_{\pi / 2}^{\pi} \frac{\sin x}{1+\cos ^{2} x} d x
$$

Answer

$\frac{\pi}{4}$

Discussion

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Video Transcript

okay. Discussion were able to solve interrogation of sine X, divided by one plus courses Square X DX. Okay. And the limit is by by two by. Okay, so first of all, we will rewrite. Our question is, it will be sign X B X divided by one Plus. It will be co sex. Holy square. Okay, so in the denominator, we have perfect spare. Okay? And now we will do the substitution process and we will take you equals two Cossacks That will give us the U equals two minus sine x dx. OK, And before substituting, we have to recommend the new parliamentary lower limit. Then the lower limit will be okay. Then the lower limit will be so lower Limit for the given X is bye bye to that is given. And we have assumed the U equals two co sex. Then you will be because access by by two. Then you will be caused by where to. Then it will be zero okay. And a parliament. Okay. And for a par limit, X equals two pi is given and U equals two Cossacks. Then it will be u equals to cause pie that is minus one Okay, So lower limit is zero and a parliament is minus one. Here. Okay. And now we will do the substitution, and it will be lower. Limit zero. Parliament minus one. Okay. And in the numerator sign X dx. OK, that is equal to minus D you okay? Minus d'you. Divided by one plus Cossacks, is you that it will be us square. Okay, now it is a simple question. We have to solve this, and we know. Do you divided by one plus you square and its integration will be our 10. You okay? It will be minus and arc, then you okay? And the limited zero to minus one. So it will be minus are 10 minus one miners, our 10 and lower limit zero. Okay, it will be minus 10 watts minus one that is minus by by four. Okay. Minus zero. Okay. And minus minus. Plus their tastes, plus by my four. And that will be our final answer. Thank you.

Vivekanand Public School
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