AP EAMCET201822 Apr 2018Morning ShiftMathematicsIndefinite IntegrationActual
d x x^3+3 x^2+2 x =
Options
- A|x|+ | x+2 x+1 |+c
- B|x|- |x+1|+ |x+2|+c
- C1 2 [ |x|+ |x+1|+ |x+2|]+c
- D1 2 ( |x^2+2 x | (x+1)^2 )+c
Correct answer
D. 1 2 ( |x^2+2 x | (x+1)^2 )+c
Step-by-step solution
Let aligned I & = d x x^3+3 x^2+2 x & = d x x (x^2+3 x+2 ) & = d x x(x+1)(x+2) aligned aligned & Let 1 x(x+1)(x+2) = A x + B x+1 + C x+2 & 1=A(x+1)(x+2)+B x(x+2)+C x(x+1) aligned Put x=0 , we get A= 1 2 Put x=-1 , we get B=-1 Put x=-2 , we get aligned & C= 1 2 & I= ( 1 2 x - 1 x+1 + 1 2(x+2) ) d x & = 1 2 x- (x+1)+ 1 2 (x+2)+C & = 1 2 [ x-2 (x+1)+ (x+2)]+C & = 1 2 | x(x+2) (x+1)^2 |+C & = 1 2 | x^2+2 x (x+1)^2 |+C . aligned