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Most Important Selected Qs for JEE AdvancedMathematicsIndefinite Integration

x^3 1+x^2 d x is equal to

Options

  1. A(1+x^2 )^ 3 2 3 - 1+x^2 +C
  2. B(1+x^2 )^ 1 / 2 (x^2+2 ) 3 +C
  3. Cx^2 1+x^2 - 2 3 (1+x^2 )^ 3 2 +C
  4. D1+x^2 ( x^2-2 3 )+c (Where C is constant of integration.)

Correct answer

D. 1+x^2 ( x^2-2 3 )+c (Where C is constant of integration.)

Step-by-step solution

Let I = x ^3 1+ x ^2 dx Put, 1+ x ^2= t aligned & I= 1 2 ( t -1 t ) dt = 1 2 ( t - 1 t ) dt = 1 2 [ 2 3 t ^ 3 2 -2 t ]+C= 1 3 (1+ x ^2 )^ 3 2 - 1+ x ^2 + C & = 1+ x ^2 [ (1+ x ^2 ) 3 -1 ]+C= 1+ x ^2 3 ( x ^2-2 )+C aligned aligned & (C) x^2 1+x^2 - 2 3 (1+x^2 )^ 3 2 +C= (1+x^2 ) 1+x^2 - 1+x^2 - 2 3 (1+x^2 )^ 3 / 2 +C & = (1+x^2 )^ 3 2 3 - 1+x^2 +C . aligned

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