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Lecture 12

# MATH 2B Lecture Notes - Lecture 12: Product RulePremium

Department
Mathematics
Course Code
MATH 2B
Professor
ERJAEE, G.
Lecture
12

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MATH 2B - Lecture 12 - Integration by Parts
Every differentiation rule has a corresponding integration rule. The integration by parts
corresponds to the product rule for differentiation.
Product Rule: [𝑓(𝑥)𝑔(𝑥)]′ = 𝑓′(𝑥)𝑔(𝑥) + 𝑓(𝑥)𝑔′(𝑥)
Integral:
∫ [𝑓(𝑥)𝑔(𝑥)]′𝑑𝑥
= 𝑓′(𝑥)𝑔(𝑥)𝑑𝑥 + 𝑓(𝑥)𝑔′(𝑥)𝑑𝑥
or 𝑓(𝑥)𝑔(𝑥) + 𝑐 = ∫ 𝑓′(𝑥)𝑔(𝑥)𝑑𝑥 + ∫ 𝑓(𝑥)𝑔′(𝑥)𝑑𝑥
Here we ignore the constant “c” on the left-hand side because the two indefinite
integrals on the right will give us another two constants later
So we have: 𝑓(𝑥)𝑔(𝑥) = ∫ 𝑓′(𝑥)𝑔(𝑥)𝑑𝑥 + ∫ 𝑓(𝑥)𝑔′(𝑥)𝑑𝑥
We rearrange to obtain:
∫ 𝑓(𝑥)𝑔′(𝑥)𝑑𝑥 = 𝑓(𝑥)𝑔(𝑥) 𝑓′(𝑥)𝑔(𝑥)𝑑𝑥
Now we simplify the formula by using new notations
Let 𝑢 = 𝑓(𝑥) and 𝑣 = 𝑔(𝑥) then 𝑑𝑢 = 𝑓′(𝑥)𝑑𝑥 𝑑𝑣 = 𝑔(𝑥)𝑑𝑥
The formula becomes
𝑢 𝑑𝑣 = 𝑢 𝑣 𝑣 𝑑𝑢
That means instead of solving 𝑢 𝑑𝑣 directly, we can use integration by parts to
arrive at the right-hand side, and solve 𝑣 𝑑𝑢 which could be simpler
Example
𝑥𝑠𝑖𝑛𝑥 𝑑𝑥
Solution:
𝑢 = 𝑥 𝑑𝑣 = 𝑠𝑖𝑛𝑥 𝑑𝑥
𝑑𝑢 = 𝑑𝑥 𝑣 = = 𝑐𝑜𝑠𝑥
𝑥𝑠𝑖𝑛𝑥 𝑑𝑥 = 𝑥(𝑐𝑜𝑠𝑥) 𝑐𝑜𝑠𝑥 𝑑𝑥
u dv u v v du
= 𝑥𝑐𝑜𝑠𝑥 + 𝑐𝑜𝑠𝑥 𝑑𝑥