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AP EAMCET201920 Apr 2019Morning ShiftMathematicsIndefinite IntegrationActual

If (I(x)= x^2( x)^2 d x ) and (I(I)=0 ), then (I(x) )

Options

  1. A( x^3 18 [8( x)^2-3 x ]+ 7 18 )
  2. B( x^3 27 [9( x)^2+6 x ]- 2 27 )
  3. C( x^3 27 [9( x)^2-6 x+2 ]- 2 27 )
  4. D( x^3 27 [9( x)^2-6 x-2 ]+ 2 27 )

Correct answer

C. ( x^3 27 [9( x)^2-6 x+2 ]- 2 27 )

Step-by-step solution

Given integral ( aligned & I(x)= x^2( x)^2 d x= x^3 3 ( x)^2- x^3 3 2 x x d x & [ by integration by parts] & = x^3 3 ( x)^2- 2 3 [ x^3 3 ( x)- x^3 3 ( 1 x ) d x ] & = x^3 3 ( x)^2- 2 3 [ x^3 3 ( x)- 1 3 x^3 3 ]+C & = x^3 27 [9( x)^2-6( x)+2 ]+C & I(1)=0 & 2 27 +C=0 C=- 2 27 & I(x)= x^3 27 [9( x)^2-6( x)+2 ]- 2 27 aligned ) Hence, option (3) is correct.

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