Question

x-a)(x-b)

Answer 1

The given function is 1 ( x−a )( x−b ) .

( x−a )( x−b )can be written as x 2 −(a+b)x+ab .(1)

Put (1) in the given function,

1 ( x−a )( x−b ) = 1 x 2 −( a+b )x+ab = 1 x−( a+b )x+ ( a+b ) 2 4 _ ( a+b ) 2 4 +ab = 1 ( x−( a+b 2 ) ) 2 − ( a−b 2 ) 2 (2)

Also, ∫ 1 x 2 − a 2 = log| x+ x 2 − a 2 |+c (3)

Now let,

x−( a+b 2 )=t dx=dt

Substitute values of t and dt in (2),

1 ( x−( a+b 2 ) ) 2 − ( a−b 2 ) 2 = 1 t 2 − ( a−b 2 ) 2 =log| t+ t 2 − ( a−b 2 ) 2 |+c =log| ( x−( a+b 2 ) )+ t 2 − ( a−b 2 ) 2 |+c =log| ( x−( a+b 2 ) )+ ( x−a )( x−b ) |+c By Using (2)

Thus, the integral of the function 1 ( x−a )( x−b ) is log| ( x−( a+b 2 ) )+ ( x−a )( x−b ) |+c.