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JEE Main2014MathematicsIndefinite IntegrationActual

The integral ^2 x ^2 x ( ^3 x+ ^3 x )^2 d x is equal to:

Options

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

Correct answer

B. .- 1 3 (1+ ^3 x . +c )

Step-by-step solution

Let I= ^2 x ^2 x ( ^3 x+ ^3 x )^2 d x aligned & I = ( x x ^3 x+ ^3 x )^2 d x & I = ( x x ^3 x (1+ ^3 x ) )^2 d x &= ( x ^2 x (1+ ^3 x ) )^2 d x & Put 1+ ^3 x= t &d t=3 ^2 x ^2 x d x or d x &= d t 3 ^2 x ^2 x & I = ^2 x ^4 x t^2 d t 3 ^2 x ^2 x aligned aligned & I = 1 3 ^2 x ^4 x t^2 d t ^2 x ^2 x ^2 x &= 1 3 ^2 x ^4 x t^2 d t ^2 x ^4 x & I = 1 3 d t t^2 = 1 3 t⁻² d t & I = 1 3 [ t⁻²⁺¹ -2+1 ]+c= -1 3 [ 1 t ]+c & or I =- 1 3 (1+ ^3 x ) +c aligned

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