Aim: How do we integrate by partial fractions ? (I)
-
Upload
joseph-rosario -
Category
Documents
-
view
27 -
download
2
description
Transcript of Aim: How do we integrate by partial fractions ? (I)
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