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What is the reverse power rule for ∫xⁿ dx?
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All Flashcards in Topic 5.11
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5.11.18 cards
What is the reverse power rule for ∫xⁿ dx?
x^(n+1)/(n+1) + C, valid for n ≠ −1 (raise the power by one, divide by the new power).
Why does an indefinite integral need + C?
Curves differing only by a constant height have the same slope, so the antiderivative is a whole family — + C names the unknown member.
What is ∫1/x dx?
ln|x| + C (the n = −1 exception to the power rule).
What is ∫eˣ dx?
eˣ + C — the exponential is its own integral.
∫1/(ax + b) dx = ?
(1/a) ln|ax + b| + C — divide by a, the derivative of the linear inside.
How does substitution work?
Let u = inside, find du = g′(x) dx, rewrite the integral fully in u, integrate, then replace u with the inside again.
Find ∫ x·e^(x²) dx.
½ e^(x²) + C (let u = x², du = 2x dx, so ∫½eᵘ du).
How do you check an integral is correct?
Differentiate your answer — it should return the original integrand.
Topic 5.11 study notes
Full notes & explanations for Integration techniques (HL only)
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