Integration by substitution
Practice Flashcards
How do you integrate (ax + b)ⁿ?
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All Flashcards in Topic 5.10
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5.10.19 cards
How do you integrate (ax + b)ⁿ?
(ax+b)ⁿ⁺¹/[a(n+1)] + C — integrate as usual, then divide by the inner coefficient a.
∫sin(ax + b) dx = ?
−cos(ax+b)/a + C.
∫cos(ax + b) dx = ?
sin(ax+b)/a + C.
∫e^(ax + b) dx = ?
e^(ax+b)/a + C.
∫1/(ax + b) dx = ?
(1/a)ln|ax+b| + C.
Why divide by the inner coefficient?
To undo the ×a that the chain rule would introduce when differentiating.
State the f'/f rule.
∫ f'(x)/f(x) dx = ln|f(x)| + C (numerator is the derivative of the denominator).
∫2x(x² + 1)³ dx = ?
(x²+1)⁴/4 + C (reverse chain: 2x is the inner derivative).
How do you check a reverse-chain integral?
Differentiate your answer — it should give back the integrand.
5.10.29 cards
What is the key idea of integration by substitution?
Let u = the inside function, replace dx using du, and integrate in u.
How do you choose u?
So that its derivative (du) already appears as a factor in the integrand.
What does du equal?
du = (du/dx) dx — used to replace the dx-part of the integrand.
After substituting, what variables should remain?
Only u (and du) — no stray x's.
For an indefinite integral, what's the last step?
Substitute back to express the answer in x (and + C).
For a definite integral by substitution, what do you do with the limits?
Convert each x-limit to a u-value, then evaluate in u.
Do you switch back to x for a definite integral?
No — once the limits are in u, evaluate directly in u.
∫2x(x²+1)³ dx by substitution u = x²+1 gives?
∫u³ du = u⁴/4 = (x²+1)⁴/4 + C.
If du = 2x dx, what is x dx?
x dx = ½ du.
Topic 5.10 study notes
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