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Integration :x+2/[(x-2)(x-3)]1/2dx
Answers
Answer:
9
√x²-5x+6 +----log{(x-5/2)+√{(x-5/2)²-1/4}
2
+c
Step-by-step explanation:
To find---->
------------
(x+2)
∫------------------------ dx
√{(x-2)(x-3)
(x+2)
=∫------------------------- dx
√(x²-2x-3x+6)
2 (x+2)
=∫------------------------- dx
2√(x²-5x+6)
2x+4
=∫-------------------------dx
2√(x²-5x+6)
( 2x-5)+5+4
=∫--------------------------dx
2√(x²-5x+6)
(2x-5)+9
=∫-------------------------dx
2√(x²-5x+6)
(2x-5) 1
=∫-------------------- dx+9 ∫--------------------dx
2√(x²-5x+6) 2√(x²-5x+6)
now for first intregation
let
x²-5x+6=t
differentiating with respect to x
( 2x -5 )dx =dt
dt 1
=∫--------------- + 9∫ ----------------------------------------
2√t 2√{(x²-5x+25/4)-25/4+6}
1 1
= ----∫t^(-1/2) dt+ 9/2∫--------------------------dx
2 √{(x-5/2)²--(1/4)
1 t^{(-1/2)+1} 9 1
=----- -------------------- +----∫--------------------------dx
2 -(1/2)+1 2 √{(x-5/2)²-(1/2)²
let x-(5/2) =u
differentiating with respect to x
dx -0 = du
dx =du
1 t^(1/2) 9 du
=--- ------------- + -----∫------------------------
2 (1/2) 2 √{u² -(1/2)²}
9
=√(x²-5x+6) +----- log{u+√{u²-(1/2)²}+c
2
9
=√(x²-5x+6)+-- log{(x-5/2)+√{(x-5/2)²-1/4}
2
+c
Hope it helps u
Thanks
see attachement also for better understanding
Answer:
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