18 flashcards · Shared on 19 August 2026 by AtomAI Library
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What is an antiderivative of a function f(x)?
A function F(x) such that F'(x) = f(x)
What is an antiderivative of a function f(x)?
A function F(x) such that F'(x) = f(x)
For n ≠ -1, which formula gives the indefinite integral of x^n?
x^(n+1)/(n+1) + C
What is ∫(1/x) dx for x ≠ 0?
ln|x| + C
Why is a constant of integration included in an indefinite integral?
Differentiation removes constants, so antiderivatives can differ by a constant
Which expression is the general antiderivative of 3x²?
x³ + C
What does the definite integral ∫ from a to b of f(x) dx represent geometrically?
The signed area between the curve and the x-axis from a to b
How does a region where f(x) is below the x-axis contribute to a definite integral?
It contributes a negative value
If F'(x) = f(x), how is ∫ from a to b of f(x) dx evaluated?
F(b) − F(a)
What happens when the limits of a definite integral are reversed?
The sign of the value changes
What is the value of ∫ from a to a of f(x) dx?
0
Which statement shows the linearity of integration?
∫[f(x)+g(x)] dx = ∫f(x) dx + ∫g(x) dx
If f(x) is above g(x) on the interval [a,b], which integral gives the area between the curves?
∫ from a to b of [f(x)−g(x)] dx
Before finding the area enclosed by two intersecting curves, what should usually be found first?
Their points of intersection
A curve crosses the x-axis within an interval. How is the total geometric area between the curve and the x-axis found?
Split the interval at each crossing and add the absolute values of the resulting integrals
What is ∫ from −1 to 1 of x dx?
0
What is the total geometric area between y = x and the x-axis from x = −1 to x = 1?
1
Which antiderivative is found most directly using the substitution u = x²?
∫2x cos(x²) dx = sin(x²) + C
If A(x) = ∫ from a to x of f(t) dt and f is continuous, what is A'(x)?
f(x)