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JEE Main202522 Jan 2025Evening ShiftMathematicsIndefinite IntegrationActual

If e ^x ( x ⁻¹ x 1-x^2 + ⁻¹ x (1-x^2 )^ 3 / 2 + x 1-x^2 ) d x= g (x)+ C , where C is the constant of integration, then g ( 1 2 ) equals :

Options

  1. A4 e 3
  2. B6 e 3
  3. C4 e 2
  4. D6 e 2

Correct answer

B. 6 e 3

Step-by-step solution

aligned & d d x ( x ⁻¹ x 1-x^2 )- ⁻¹ x ( 1 1-x^2 + x 2 x 2 1-x^2 1-x^2 ) & = x 1-x^2 1 1-x^2 & = ⁻¹ x (1-x^2 )^ 3 / 2 + x 1-x^2 aligned aligned & Hence, I= e^x (f(x)+f^ (x) ) d x & =e^x f(x)+C & I=e^x x ⁻¹ x 1-x^2 +C=g(x)+C & g(x)= x e^x ⁻¹ x 1-x^2 and g(1 / 2)= 6 e 3 aligned

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