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Flip to reveal answersWhat is the key idea of integration by substitution?
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All 9 Flashcards — Substitution
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Question
What is the key idea of integration by substitution?
Answer
Let u = the inside function, replace dx using du, and integrate in u.
Question
How do you choose u?
Answer
So that its derivative (du) already appears as a factor in the integrand.
Question
What does du equal?
Answer
du = (du/dx) dx — used to replace the dx-part of the integrand.
Question
After substituting, what variables should remain?
Answer
Only u (and du) — no stray x's.
Question
For an indefinite integral, what's the last step?
Answer
Substitute back to express the answer in x (and + C).
Question
For a definite integral by substitution, what do you do with the limits?
Answer
Convert each x-limit to a u-value, then evaluate in u.
Question
Do you switch back to x for a definite integral?
Answer
No — once the limits are in u, evaluate directly in u.
Question
∫2x(x²+1)³ dx by substitution u = x²+1 gives?
Answer
∫u³ du = u⁴/4 = (x²+1)⁴/4 + C.
Question
If du = 2x dx, what is x dx?
Answer
x dx = ½ du.
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Topic 5.10 hub
Integration by substitution
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