MHT CET202523 Apr 2025Morning ShiftMathematicsIndefinite IntegrationActual
( x-3 x^2+9 )^2 ~d x=
Options
- A1 3 ⁻¹ ( x 3 )- 3 x^2+9 +c , where c is the constant of integration.
- B1 3 ⁻¹ ( x 3 )- 1 x^2+9 +c , where c is the constant of integration.
- C1 3 ⁻¹ ( x 3 )+ 3 x^2+9 +c , where c is the constant of integration.
- D1 3 ⁻¹ ( x 3 )- 1 x^2+9 +c , where c is the constant of integration.
Correct answer
C. 1 3 ⁻¹ ( x 3 )+ 3 x^2+9 +c , where c is the constant of integration.
Step-by-step solution
Evaluate the integral ( x-3 x^2+9 )^2 d x . Expand the integrand and rewrite the numerator: ( x-3 x^2+9 )^2 = x^2 - 6x + 9 (x^2+9)^2 = (x^2+9) - 6x (x^2+9)^2 = 1 x^2+9 - 6x (x^2+9)^2 . The integral becomes: I = 1 x^2+9 d x - 6x (x^2+9)^2 d x . The first integral is standard: 1 x^2+9 d x = 1 3 ⁻¹ ( x 3 ) . For the second, use substitution with u = x^2+9 , d u = 2x d x : 6x (x^2+9)^2 d x = 3 u⁻² d u = - 3 u = - 3 x^2+9 . Combining results: I = 1 3 ⁻¹ ( x 3 ) + 3 x^2+9 + C . This matches option C.