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MHT CET202210 Aug 2022Morning ShiftMathematicsIndefinite IntegrationActual

The integral ^2 x ^2 x ( ^5 x+ ^3 x ^2 x+ ^3 x ^2 x+ ^5 x )^2 d x is equal to (Where C is a constant of integration).

Options

  1. A1 1+ ^3 x +C
  2. B-1 1+ ^3 x +C
  3. C1 3 (1+ ^3 x ) +C
  4. D-1 3 (1+ ^3 x ) +C

Correct answer

D. -1 3 (1+ ^3 x ) +C

Step-by-step solution

aligned & ^2 x ^2 x ( ^5 x+ ^3 x ^2 x+ ^3 x ^2 x+ ^5 x )^2 d x & = ^2 x ^6 x d x ( ^5 x+ ^2 x+ ^3 x+1 )^2 [dividing N^r and D^r by ¹⁰ x ] & = ^2 x ( ^2 x+1 )^2 ^2 x d x ( ^2 x+1 )^2 ( ^3 x+1 )^2 & = ^2 x ^2 x d x ( ^3 x+1 )^2 aligned aligned & = 1 3 d t t^2 & Let ^3 x+1=t i.e., 3 ^2 x ^2 x ~d x= d t & =- 1 3 t +c & = -1 3 ( ^3 x+1 ) +c aligned

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