Let f ( t ) = ( 1 - sin ( log e t ) 1 - cos ( log e t ) ) d t , t 1 . If f ( e / 2 ) = - e / 2 and f ( e / 4 ) = e / 4 , then equals [2026]

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Solution

Q.
Let $f (t)$ $= \int (\frac{1 - \sin (\log_{e} t)}{1 - \cos (\log_{e} t)}) d t, t > 1 .$

If $f (e^{π / 2}) = - e^{π / 2} and f (e^{π / 4}) = α e^{π / 4},$ then $α$ equals [2026]

1 $- 1 - \sqrt{2}$

2 $1 + \sqrt{2}$

3 $- 1 - 2 \sqrt{2}$

4 $- 1 + \sqrt{2}$

Ans.
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