Question

integrate {x-1 / (x-2)(x-3) }dx​

Answer 1

EXPLANATION.

∫ ( x - 1 ) / ( x - 2)(x - 3 )dx.

we can use the concept of integration by partial fraction.

NOTE = partial fraction is used only when Coefficient of Denominator > Numerator,

⇒ ( x - 1 ) / ( x - 2 )( x - 3 ) = A/( x - 2 ) + B/( x - 3 ).

⇒ ( x - 1 ) = A ( x - 3 ) + B ( x - 2 ).

put the value of x = 3 in equation,

⇒ ( 3 - 1 ) = A ( 3 - 3 ) + B ( 3 - 2 ).

⇒ 2 = 0 + B.

⇒ B = 2.

Put the value of x = 2 in equation.

⇒ ( 2 - 1 ) = A ( 2 - 3 ) + B ( 2 - 2 ).

⇒ 1 = A(-1) + 0.

⇒ A = -1.

Put the value of A and B in equation,

⇒ ∫ -1/(x - 3 )dx + ∫2/( x - 2 )dx.

⇒ -㏑( x - 3 ) + 2㏑( x - 2 ) + c.

                                             

MORE INFORMATION.

Integration of trigonometric function.

TYPE = 1.

(1) = ∫ dx/a + bsin²x.

(2) = ∫ dx/a + bcos²x.

(3) = ∫ dx/acos²x + b sin(x). cos(x) + csin²x.

(4) = ∫ dx/( a sin(x) + b cos(x) )²

Divide Numerator and Denominator by cos²x in all such type of integrals and then put ⇒ tan(x) = t.