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Flip to reveal answersState the integration by parts formula.
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All 8 Flashcards — Integration by parts
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Question
State the integration by parts formula.
Answer
∫ u dv = uv − ∫ v du. You differentiate u and integrate dv.
Question
What does LIATE help you decide?
Answer
Which factor to call u: Log, Inverse-trig, Algebra (powers of x), Trig, Exponential — first listed becomes u.
Question
When do you use integration by parts (not substitution)?
Answer
For a PRODUCT of two different kinds of function (e.g. x·eˣ, x·sin x, x·ln x) where no inner-derivative pair is present.
Question
Find ∫ x cos x dx.
Answer
u = x, dv = cos x dx → x sin x − ∫ sin x dx = x sin x + cos x + C.
Question
How is ∫ ln x dx found by parts?
Answer
u = ln x, dv = dx (v = x): x ln x − ∫ 1 dx = x ln x − x + C.
Question
How many times must you apply parts for ∫ x² eˣ dx?
Answer
Twice — each round drops the power of x by one (x² → x → constant).
Question
Find ∫ x ln x dx.
Answer
u = ln x, dv = x dx → (x²/2)ln x − ∫ x/2 dx = (x²/2)ln x − x²/4 + C.
Question
Find ∫ x² eˣ dx.
Answer
Parts twice: eˣ(x² − 2x + 2) + C.
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Full study notes for Integration by parts
Topic 5.16 hub
Integration by parts (HL only)
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