MHT CET202522 Apr 2025Evening ShiftMathematicsIndefinite IntegrationActual
x^3 x^4+5 x^2+4 ~d x=
Options
- A1 3 ( (x^2+4 )^2 x^2+1 )+c , where c is the constant of integration
- B( (x^2+4 )^2 x^2+1 )+c, where c is the constant of integration
- C3log ( (x^2+4 )^2 x^2+1 )+c, where c is the constant of integration
- D2 3 ( (x^2+4 )^2 x^2+1 )+c , where c is the constant of integration
Correct answer
A. 1 3 ( (x^2+4 )^2 x^2+1 )+c , where c is the constant of integration
Step-by-step solution
Let I = x^3 x^4+5x^2+4 d x . Substitute u = x^2 , so d u = 2x , d x and x , d x = 1 2 d u . The integral becomes I = 1 2 u u^2+5u+4 d u . Factor the denominator: u^2+5u+4 = (u+1)(u+4) . Decompose the rational function using partial fractions: u (u+1)(u+4) = A u+1 + B u+4 . Solving for constants: u = A(u+4) + B(u+1) . Setting u = -1 gives A = - 1 3 , and u = -4 gives B = 4 3 . The integral simplifies to I = 1 2 (- 1 3 1 u+1 d u + 4 3 1 u+4 d u ) . Integrating yields I = - 1 6 |u+1| + 2 3 |u+4| + C . Substitute back