Aim: How do we integrate by partial fractions? (I)

Do Now: Write the partial fractions decomposition of

𝐴𝑥−1

+𝐵𝑥+2

=𝐴 (𝑥+2 )+𝐵(𝑥− 1)

(𝑥−1)(𝑥+2)𝐴𝑥+2𝐴+𝐵𝑥−𝐵=𝑥+5

( 𝐴+𝐵 )𝑥+(2 𝐴−𝐵 )=𝑥+5

𝑨=𝟐 ,𝑩=−𝟏

∫ 𝑥+5

𝑥2+𝑥− 2𝑑𝑥

¿∫( 2𝑥−1

−1

𝑥+2 )𝑑𝑥¿2 𝑙𝑛|𝑥− 1|−𝑙𝑛|𝑥+2|+𝐶

𝑢=𝑥−1 ,𝑑𝑢=𝑑𝑥

𝑢=𝑥+2,𝑑𝑢=𝑑𝑥

A rational function whose numerator has higher degree than the denominator, divide and write the form of

∫ 𝑥3+𝑥𝑥− 1

𝑑𝑥¿∫(𝑥2+𝑥+2+2

𝑥−1)𝑑𝑥

¿ 𝑥3

3+𝑥

2

2+2𝑥+2 𝑙𝑛|𝑥− 1|+𝐶

∫ 𝑥2+2 𝑥−12 𝑥3+3𝑥2 −2𝑥

𝑑𝑥

¿∫ 12

1𝑥

+15

12𝑥− 1

+110

1𝑥+2

𝑑𝑥

¿12𝑙𝑛|𝑥|+ 1

10𝑙𝑛|2 𝑥−1|− 1

10𝑙𝑛|𝑥+2|+𝐶

In integrating the middle term we have made the mental substitution u = 2x – 1, which gives du = 2dx and

∫ 𝑑𝑥𝑥2−𝑎2

¿ 12𝑎∫( 1

𝑥−𝑎−

1𝑥+𝑎 )𝑑𝑥

𝑨=𝟏𝟐𝒂

,𝑩=−𝟏𝟐𝒂

¿1

2𝑎(𝑙𝑛|𝑥−𝑎|−𝑙𝑛|𝑥+𝑎|)+𝐶

¿ 12𝑎

𝑙𝑛|𝑥−𝑎𝑥+𝑎|+𝐶

∫ 𝑥4 −2𝑥2+4 𝑥+1𝑥3−𝑥2 −𝑥+1

𝑑𝑥¿ 𝒙+𝟏+𝟒 𝒙

𝒙𝟑−𝒙𝟐− 𝒙+𝟏

𝒙𝟒−𝟐𝒙𝟐+𝟒 𝒙+𝟏𝒙𝟑− 𝒙𝟐−𝒙+𝟏

¿∫ [𝑥+1+ 1𝑥−1

+ 2

(𝑥− 1)2−

1𝑥+1 ]𝑑𝑥  

¿ 𝑥2

2+𝑥+ 𝑙𝑛|𝑥−1|− 2

𝑥−1−𝑙𝑛|𝑥+1|+𝐶

¿ 𝑥2

2+𝑥−

2𝑥−1

+𝑙𝑛|𝑥−1𝑥+1 |+𝐶