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MHT CET20244 May 2024Evening ShiftMathematicsIndefinite IntegrationActual

The value of d x x^2 (x^4+1 )^ 3 4 is

Options

  1. A(x^4+1 )^ 1 4 + c , where c is a constant of integration.
  2. B(x^4+1 )^ 1 4 x + c , where c is a constant of integration.
  3. C- (x^4+1 )^ 1 4 x + c , where c is a constant of integration.
  4. D- (x^4+1 )^ 1 4 + c , where c is a constant of integration.

Correct answer

C. - (x^4+1 )^ 1 4 x + c , where c is a constant of integration.

Step-by-step solution

Let aligned I & = d x x^2 (x^4+1 )^ 3 4 & = d x x^2 [x^4 (1+ 1 x^4 ) ]^ 3 4 & = d x x^5 (1+ 1 x^4 )^ 3 4 & = (1+ 1 x^4 )^ -3 4 x^5 aligned aligned & Let 1+ 1 x^4 = t & -4 x^5 ~d x= dt & d x x^5 =- dt 4 & I = 1 4 t ^ -3 4 -3 4 dt & = -1 4 t^ -3 4 +1 -3 4 +1 & =-t^ 1 4 + c aligned aligned I & =- (1+ 1 x^4 )^ 1 4 + c & =- ( x^4+1 x^4 )^ 1 4 + c I & = - (x^4+1 )^ 1 4 x + c aligned

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