Skeletal Notes-8.2 Integration by Parts.pdf - 8.2 Integration by Parts Objectives 1 Integration by Parts 2 Column integration Quick Review the Product

Skeletal Notes-8.2 Integration by Parts.pdf - 8.2...

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8.2 Integration by Parts Objectives: 1. Integration by Parts 2. Column integration Quick Review: the Product Rule If ?(?) ??? ?(?) are both differentiable and ?(?) = ?(?)?(?) , then ? ?? ?(?) = ? ?? [?(?)?(?)] = ?(?) ? ?? [?(?)] + ?(?) ? ?? [?(?)] Write the rule in prime notation: Example 1 : ? = (? 2 + ? + 3)(? x − 5) Find ?? ?? Simplify the result. Algebra Review : ? ? = ? ∙ ? −1 , ??𝑟 ? ≠ 0 Example 2 : Rewrite each as a product: ? = ln ? ? ? = ? 2 ? 3𝑥 ? = 5 ? 6 ? = 1 √? 3
(2) Integration by Parts Integration by Parts ∫ ?(𝒙)?′(𝒙) 𝒅𝒙 = ?(𝒙)?(𝒙) − ∫ ?(𝒙)?′(𝒙)𝒅𝒙 Or letting ? = ?(?) and ? = ?(?), we get ∫ ?𝒅? = ?? − ∫ ?𝒅? Tips: 1. Choose ? = ?(?) to be a function that becomes simpler when differentiated and ?? = ?′(?)?? to be the part that can be readily integrated to give back v. 2. Check v by differentiating it before using the formula. Experiences suggest the following rule: whichever function comes first in the list LATE is to be u and the rest is dv. L : Logarithmic functions: ln ?, log 2 ? , etc A : Algebraic functions: polynomials ? 2 , 3? 7 , ? 2 + 6 , etc T : Trigonometric functions: sin ? , cos ? , etc E : Exponential function: ? ? , 13 ? , etc Example 3 : For each function, what will you choose for u? What will be dv?

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