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AP EAMCET201920 Apr 2019Morning ShiftMathematicsDefinite IntegrationActual

( ₀^1 _e(1+x) 1+x^2 d x= )

Options

  1. A( 4 ₆ 2 )
  2. B( 6 ₆ 6 )
  3. C( 2 ₆ 8 )
  4. D( 8 ₆ 2 )

Correct answer

D. ( 8 ₆ 2 )

Step-by-step solution

Given integral (I= ₀^1 _e(1+x) 1+x^2 d x ) put (x= ), so at (x=0, =0 ) and at (x=1, = 4 ) and (d x= ^2 d ) Now, (I= ₀^ 4 _e(1+ ) ^2 ^2 d ) (= ₀^ 4 _e(1+ ) d ) ...(i) On, applying property, ( ₀^a f(x) d x= ₀^a f(a-x) d x ), we get ( aligned & I= ₀^ 4 _e (1+ ( 4 - ) ) d & = ₀^ 4 _e (1+ 1- 1+ ) d & = ₀^ 4 _e ( 2 1+ ) d & = ₀^ 4 _e(2) d x- ₀^ 4 _e(1+ ) d & I= 4 _e(2)-I [from Eq. (i)] & 2 I= 4 _e(2) I= 8 _e(2) aligned ) Hence, option (4) is correct.

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