Partial Fractions

partial fractions

Algebra for polynomial fractions. This is expected knowledge.

= \frac{x}{2 x - 1} - \frac{3}{x + 4} = \frac{x ( x + 4 )}{( 2 x - 1 ) ( x + 4 )} - \frac{3 ( 2 x - 1 )}{( 2 x - 1 ) ( x + 4 )} = \frac{x ^{2} + 4 x}{2 x ^{2} + 7 x - 4} - \frac{6 x - 3}{2 x ^{2} + 7 x - 4} = \frac{x ^{2} + 4 x - 6 x - 3}{2 x ^{2} + 7 x - 4}

We can also reverse this through a process called “partial fracti ons”.

A + 3 B 2 A - 5 B 2 A + 6 B 2 A - 5 B subtract the previous two functions 0 - 11 B B 3 (- \frac{1}{11}) + A = 1 A = \frac{x + 3}{3 x ^{2} + x - 10} = \frac{x + 3}{( 3 x - 5 ) ( x + 2 )} = \frac{A}{3 x - 5} + \frac{B}{x + 2} = \frac{A ( x + 2 ) + B ( 3 x - 5 )}{( 3 x - 5 ) ( x + 2 )} = \frac{( A + 3 B ) x + 2 A - 5 B}{( 3 x - 5 ) ( x + 2 )} = 1 = 3 = 2 = 3 = - 1 = - \frac{1}{11} = \frac{14}{11} = \frac{\frac{14}{11}}{3 x - 5} + \frac{- \frac{1}{11}}{x + 2} We know from the first fraction multiply by 2 so things can cancel

This is useful for finding integrals of complicated fractions.

\int \frac{x + 3}{3 x ^{2} + x - 10} d x = \int \frac{\frac{14}{11}}{3 x - 5} - \int \frac{\frac{1}{11}}{x + 2} = \frac{14}{11} \int \frac{1}{3 x - 5} - \frac{1}{11} \int \frac{1}{x + 2} = \frac{14}{33} l n ∣3 x - 5∣ - \frac{1}{11} l n ∣ x + 2∣ + C

\int \frac{x - 2}{x ^{2} + 5 x} d x A + B 5 A A B = \frac{x - 2}{x ( x + 5 )} = \frac{A}{x} + \frac{B}{x + 5} = \frac{A ( x + 5 )}{x ( x + 5 )} + \frac{B ( x )}{x ( x + 5 )} = \frac{( A + B ) x + 5 A}{x ( x + 5 )} = 1 = - 2 = - \frac{2}{5} = \frac{7}{5} = \frac{- \frac{2}{5}}{x} + \frac{\frac{7}{5}}{x + 5} = \int \frac{- \frac{2}{5}}{x} + \int \frac{\frac{7}{5}}{x + 5} = - \frac{2}{5} \int \frac{1}{x} + \frac{7}{5} \int \frac{1}{x + 5} = - \frac{2}{5} l n ∣ x ∣ + \frac{7}{5} l n ∣ x ∣ + C

Section 6.3 (pg.303): #5-12, #19-28 Section 6.4 (pg.312): #5-26, #27-32 Section 6.5 (pg.320): #7-12, 16, 17, 18, 20

\int \frac{7 x + 7}{x ^{2} + 3 x - 10} \frac{A}{( x - 2 )} + \frac{B}{( x + 5 )} \frac{A ( x + 5 )}{( x - 2 ) ( x + 5 )} + \frac{B ( x - 2 )}{( x - 2 ) ( x + 5 )} A + B = 7 - A 2 + B 5 = 7 A 5 + B 5 = 35 - A 7 = - 28 A = 4 B = 3 \int \frac{4}{x - 2} + \int \frac{3}{x + 5} 4 l n ∣ x ∣ + 3 l n ∣ x ∣ + C subtract two funcs above

\int \frac{7 x - 2}{x ^{2} + x} A (x + 1), B (x) A + B = 7 A = - 2 B = 9 \int \frac{- 2}{x} + \frac{9}{x + 1} - 2 l n ∣ x ∣ + 9 l n ∣ x ∣ = \frac{A}{x} + \frac{B}{x + 1}