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 ? ≠ 0Example 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)𝒅𝒙Or letting ? = ?(?)and? = ?(?),weget∫ ?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 ?,log2?, etcA: Algebraic functions: polynomials ?2,3?7, ?2+ 6, etcT: Trigonometric functions: sin ? ,cos ?, etcE: Exponential function: ??,13?, etcExample 3: For each function, what will you choose for u? What will be dv?