Question

निम्नलिखित प्रश्न में परिमेय फलन का समाकलन कीजिए:

`x/((x + 1)(x + 2))`

Answer 1

माना `x/((x + 1)(x + 2)) = A/(x + 1) + B/(x + 2)`

`=> x/((x + 1)(x + 2)) = (A(x + 2) + B(x + 1))/((x + 1)(x + 2))`

x = -1 रखने पर,

-1 = A (-1 + 2) ⇒ -1 = A

⇒  A = -1

x = -2 रखने पर,

-2 = B (-2 + 1) ⇒ -2 = -B

⇒  B = 2

∴ `x/ ((x + 1) (x + 2)) = (-1)/ (x + 1) + 2/ (x + 2)`

∴ `I = int x/ ((x + 1) (x + 2)) dx`

`= int [(-1)/ (x + 1) + 2/ (x + 2)] dx`

`= int (-1)/ ((x + 1))  dx + int 2/ (x + 2)  dx`

= - log |x + 1| + 2 log |x + 2| + C

= - log |x + 1| + log |x + 2|² + C

`= log |((x + 2)^2)/(x + 1)| + C`