Integration techniques
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Flip to reveal answersWhat is the reverse power rule for ∫xⁿ dx?
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All 8 Flashcards — Integration techniques
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Question
What is the reverse power rule for ∫xⁿ dx?
Answer
x^(n+1)/(n+1) + C, valid for n ≠ −1 (raise the power by one, divide by the new power).
Question
Why does an indefinite integral need + C?
Answer
Curves differing only by a constant height have the same slope, so the antiderivative is a whole family — + C names the unknown member.
Question
What is ∫1/x dx?
Answer
ln|x| + C (the n = −1 exception to the power rule).
Question
What is ∫eˣ dx?
Answer
eˣ + C — the exponential is its own integral.
Question
∫1/(ax + b) dx = ?
Answer
(1/a) ln|ax + b| + C — divide by a, the derivative of the linear inside.
Question
How does substitution work?
Answer
Let u = inside, find du = g′(x) dx, rewrite the integral fully in u, integrate, then replace u with the inside again.
Question
Find ∫ x·e^(x²) dx.
Answer
½ e^(x²) + C (let u = x², du = 2x dx, so ∫½eᵘ du).
Question
How do you check an integral is correct?
Answer
Differentiate your answer — it should return the original integrand.
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Integration techniques (HL only)
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