Basic Integration
∫ (4x³ − 6x) dx
x⁴ − 3x² + C
∫₂₀ (3x²) dx
8
If A(x) = ∫ˣ₀ t² dt, find A′(x).
x²
If f(x) is positive on [1,5], what can be said about ∫⁵₁ f(x) dx?
It is positive.
Velocity is the rate of change of what quantity?
Position
∫ (3x² + 4x − 7) dx
x³ + 2x² − 7x + C
∫₄₁ (2x − 1) dx
12
If A(x) = ∫ˣ₁ (4t + 3) dt, find A′(x).
4x + 3
If f(x) is negative on [2,6], what can be said about ∫⁶₂ f(x) dx?
It is negative.
If velocity is measured in miles per hour, integrating velocity gives what quantity?
Change in position (displacement).
∫ (2/x + x⁴) dx
2ln|x| + x⁵/5 + C
∫₃₀ (x² + 2x) dx
18
If A(x) = ∫ˣ²₀ cos(t) dt, find A′(x).
2x cos(x²)
If ∫⁴₁ f(x) dx = 8 and ∫⁷₄ f(x) dx = 5, find ∫⁷₁ f(x) dx.
13
A car travels at 50 mph for 2 hours. What is the accumulated distance?
100 miles
Find the most general antiderivative of f′(x) = 6x² − 8.
2x³ − 8x + C
∫₂₋₂ (x³ + 5)
20
If A(x) = ∫³ˣ (t² − 1) dt, find A′(2).
3
If ∫⁵₂ f(x) dx = 10, find ∫²₅ f(x) dx.
−10
A tank fills at 8 gallons per minute for 10 minutes. How much water accumulates?
80 gallons
If F′(x) = 3x² − 4x + 1 and F(0) = 7, find F(x).
F(x) = x³ − 2x² + x + 7
∫₁₀ (4x³ − 2x² + 6) dx
20/3
If A(x) = ∫⁵ˣ³ (t² + 4) dt, find A′(x).
3x²(x⁶ + 4)
If ∫⁴₀ f(x) dx = 7 and ∫⁶₄ f(x) dx = −3, find ∫⁶₀ f(x) dx.
4
The rate at which a population grows is 200 people per year for 5 years. What is the total change in population?
1000 people