Introduction to integration
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What is integration?
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All Flashcards in Topic 5.5
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5.5.19 cards
What is integration?
The reverse of differentiation (antidifferentiation).
State the rule for ∫xⁿ dx.
xⁿ⁺¹/(n+1) + C, for n ≠ −1.
Why must you add + C to an indefinite integral?
Differentiating any constant gives 0, so the original could have had any constant.
How do you integrate a constant like 5?
It becomes 5x (5 = 5x⁰, add 1 to the power).
How do you integrate a polynomial?
Integrate each term with the power rule, then add a single + C.
∫√x dx = ?
∫x^(1/2) dx = (2/3)x^(3/2) + C.
∫1/x² dx = ?
∫x⁻² dx = −1/x + C.
How do you find f(x) from f'(x) and a point?
Integrate f'(x) (with + C), then substitute the point to find C.
Which power can't you integrate with this rule?
n = −1 (∫x⁻¹ dx = ln|x| + C, a special case).
5.5.29 cards
What does a definite integral ∫ₐᵇ f(x) dx represent (f ≥ 0)?
The area between the curve and the x-axis from x = a to x = b.
How do you evaluate a definite integral?
Integrate to get F(x), then compute F(b) − F(a).
Does a definite integral need + C?
No — the constant cancels in F(b) − F(a).
What is F(b) − F(a) in words?
The antiderivative at the top limit minus the antiderivative at the bottom limit.
What happens if you swap the limits?
The sign of the integral flips.
How do you find an unknown limit from a given area?
Set the definite integral equal to the area and solve for the limit.
∫₁³ 2x dx = ?
[x²]₁³ = 9 − 1 = 8.
∫₀² x² dx = ?
[x³/3]₀² = 8/3.
Indefinite vs definite integral?
Indefinite gives a function + C; definite (with limits) gives a number.
Topic 5.5 study notes
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