Integration by partial fraction.

∫ ( 5x + 4 )/( x + 2 ) ( x - 4 ) dx 

Let's try solving this particular problem using partial fraction method of integration.

In partial fraction, we have to rewrite this function in the form [A/( x + 2 ) + B/( x - 4 )]

and then split each rational expression in to two independent functions and add them after finding the values of A and B.

( 5x + 4 ) / ( x + 2 ) ( x - 4 ) = A / ( x + 2 ) + B / ( x - 4 )

ie  A ( x - 4 ) + B ( x + 2 ) / ( x + 2 ) ( x - 4 ) = ( 5x + 4 )/ ( x + 2 ) ( x - 4 )

Now equating numerators of both sides

A ( x - 4 ) + B ( x + 2 ) = ( 5x + 4 )

Ax - 4A + Bx + 2B = 5x + 4

x ( A + B ) +( 2B - 4A ) = 5x + 4

Equating coefficient of x on both sides A + B = 5 , ∴ A = 5 - B-------------(1)

Equating constants on both sides 2B - 4A = 4------------------------------------(2)

Solving (1) and (2) 2B - 4( 5 - B ) = 4

2B - 20 + 4B = 4,    ∴ 6 B = 24 or B = 4 and A = 5 - B or A = 1

∴ ∫ (5x + 4 ) / ( x + 2 ) ( x - 4 ) dx = ∫ ( 1 / ( x + 2 ) + 4 / ( x - 4 ) dx 

= ∫ 1 / ( x + 2 ) dx + ∫ 4 / ( x - 4 ) dx

= ∫ 1/( x + 2 ) dx + 4 ∫ 1/( x - 4 ) dx

= ln | x + 2 | + 4 ln | x - 4 | + C