MHT CET202615 April 2026Evening ShiftMathematicsIndefinite IntegrationActual
x^2 , d x (x^2 + 2)(x^2 + 5) =
Options
- A- ( 2 3 ⁻¹ x 2 + 5 3 ⁻¹ x 5 ) + c , where c is the constant of integration
- B( 2 3 ⁻¹ x 2 + 5 3 ⁻¹ x 5 ) + c , where c is the constant of integration
- C2 3 ⁻¹ ( x 2 ) - 5 3 ⁻¹ ( x 5 ) + c , where c is the constant of integration
- D- 2 3 ⁻¹ ( x 2 ) + 5 3 ⁻¹ ( x 5 ) + c , where c is the constant of integration
Correct answer
D. - 2 3 ⁻¹ ( x 2 ) + 5 3 ⁻¹ ( x 5 ) + c , where c is the constant of integration
Step-by-step solution
Let I = x^2 (x^2 + 2)(x^2 + 5) , d x To resolve into partial fractions, substitute x^2 = t : t (t + 2)(t + 5) = A t + 2 + B t + 5 t = A(t + 5) + B(t + 2) Putting t = -2 , we get -2 = 3A A = - 2 3 Putting t = -5 , we get -5 = -3B B = 5 3 Therefore, x^2 (x^2 + 2)(x^2 + 5) = - 2 3(x^2 + 2) + 5 3(x^2 + 5) Integrating with respect to x : I = - 2 3 1 x^2 + ( 2 )^2 , d x + 5 3 1 x^2 + ( 5 )^2 , d x I = - 2 3 ( 1 2 ⁻¹ x 2 ) + 5 3 ( 1 5 ⁻¹ x 5 ) + c I = - 2 3 ⁻¹ ( x 2 ) + 5 3 ⁻¹ ( x 5 ) + c Answer: - 2 3 ⁻¹ ( x 2 ) + 5 3 ⁻¹