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Integrals

Question
CBSEENMA12036231

Let f (x) be a non−negative continuous function such that the area bounded by the curve y = f (x), x−axis and the ordinates x = π/4  and x = β >  π/4 open parentheses straight beta space sin space straight beta space plus space straight pi over 4 space cos space straight beta space plus space square root of 2 straight beta close parentheses. Then f (π/2) is

  • open parentheses straight pi over 4 plus square root of 2 minus 1 close parentheses
  • open parentheses straight pi over 4 minus square root of 2 plus 1 close parentheses
  • open parentheses 1 minus straight pi over 4 minus square root of 2 plus 1 close parentheses
  • open parentheses 1 minus space straight pi over 4 plus space square root of 2 close parentheses

Solution

D.

open parentheses 1 minus space straight pi over 4 plus space square root of 2 close parentheses

Given that 
integral subscript straight pi divided by 4 end subscript superscript straight beta space straight f left parenthesis straight x right parenthesis space dx space equals space straight beta space sin space straight beta space plus space straight pi over 4 space cos space straight beta space plus space square root of 2 straight beta
Differentiating space. straight w. straight r. space straight t space straight beta
straight f left parenthesis straight beta right parenthesis space equals space straight beta space cos space straight beta space plus space sin space straight beta space minus space straight pi over 4 space sin space straight beta space plus space square root of 2
space straight f open parentheses straight pi over 2 close parentheses space equals space open parentheses 1 minus straight pi over 4 close parentheses space sin space straight pi over 2 space plus space square root of 2 space equals space 1 minus straight pi over 4 space plus space square root of 2

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