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AP EAMCET20226 Jul 2022Evening ShiftMathematicsDefinite IntegrationActual

Assertion (A) ₂^e ( 1 _e x - 1 ( _e x )^2 ) d x=e-2 ₂ e Reason (R) _a^b e^x (f(x)+f^ (x) ) d x=e^b f(b)-e^a f(a)

Options

  1. AA and R are true, R is the correct explanation to A .
  2. BA and R are false, R is not the correct explanation to A .
  3. CA is true and R is false, R is not the correct explanation to A .
  4. DA is false and R is true, R is not the correct explanation to A .

Correct answer

A. A and R are true, R is the correct explanation to A .

Step-by-step solution

Assertion Let I= ₂^e ( 1 _e x - 1 ( _e x )^2 ) d x Let _e x=y aligned x & =e^y d x & =e^y d y I & = _ _e 2 ^1 e^y [ 1 y + ( -1 y^2 ) ] d y aligned [Using the formula _a^b e^x [f(x)+f^ (x) ] d x= [e^x f(x) ]_a^b ] aligned I & = [e^y 1 y ]_ - e^2 ^1 & =e- 2 _e 2 =e-2 ₂ e aligned Hence, Assertion and Reason both are true and Reason is the correct explanation of Assertion.

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