AP EAMCET201923 Apr 2019Morning ShiftMathematicsDefinite IntegrationActual
( ₀^ 2 ^2 x x+ x d x= )
Options
- A( 3 2 ( 2 +1)^ 1 2 )
- B( 1 2 ( 2 +1) )
- C( 2 3 ( 3 +1) )
- D( 2 3 ( 2 -1) )
Correct answer
B. ( 1 2 ( 2 +1) )
Step-by-step solution
Let (I= ₀^ 2 ^2 x x+ x d x ) ...(i) On applying property ( ₀^a f(x) d x= ₀^a f(a-x) d x ), we get, (I= ₀^ 2 ^2 x x+ x d x ) ...(ii) On adding Eqs. (i) and (ii), we get ( aligned 2 I & = ₀^ 2 ^2 x+ ^2 x x+ x d x & = ₀^ 2 d x x+ x & = 1 2 ₀^ 2 cosec ( 4 +x ) d x & = 1 2 [ | cosec ( 4 +x )- ( 4 +x ) | ]₀^ / 2 I & = 1 2 2 [ ( 2 +1)- ( 2 -1)] & = 1 2 2 [2 ( 2 +1)]= 1 2 ( 2 +1) aligned ) Hence, option (b) is correct.