Manipal MET2013MathematicsIndefinite Integration
If 1 x(x+1)(x+2) (x+n) = A₀ x + A₁ x+1 + A₂ x+2 + + A_n x+n then A_r is equal to
Options
- Ar!(1)^r (n-r)!
- B(-1)^2 r!(n-r)!
- C1 r!(n-r)!
- DNone of these
Correct answer
D. None of these
Step-by-step solution
We have, aligned & 1 x(x+1)(x+2) (x+n) = A₀ x + A₁ x+1 + + & A_t x+r + + A_n x+n aligned aligned & 1=A₀(x+1)(x+2) (x+n) & + +A_r x(x+1)(x+2) (x+r-1)(x+r+1) & (x+n)+ +A_n x(x+1) (x+n-1) aligned Putting x=-r , we get; array r 1=A_r(-r)(-r+1)(-r+2) (-3)(-2)(-1)(1)(2) (n-r) array aligned 1=A_r(-1)^r r(r-1)(r-2) (3 2 1 (1 2 3 (n-r)) aligned A_r=(-1)^r / r!(n-r)!