08 Antiderivatives and Indefinite Integrals Free Online FlashCards

What condition defines an antiderivative F of f?

F is an antiderivative of f on an interval when F′(x) = f(x).

What does an indefinite integral represent?

∫ f(x) dx = F(x) + C, where F′(x) = f(x). It represents the entire family of antiderivatives.

State the power rule for integration.

For n ≠ −1, ∫ xⁿ dx = xⁿ⁺¹/(n + 1) + C.

What is the antiderivative of 1/x?

∫ 1/x dx = ln|x| + C.

Find all antiderivatives of 6x − 4.

∫(6x − 4) dx = 3x² − 4x + C.

Integrate 5x⁴ − 2x³ + 7x − 9.

∫(5x⁴ − 2x³ + 7x − 9) dx = x⁵ − x⁴/2 + 7x²/2 − 9x + C.

What does an initial-value problem determine?

An initial-value problem combines a differential equation with a condition such as y(x₀) = y₀. The condition determines the integration constant.

Solve dy/dx = 4x³ − 2x with y(1) = 5.

Integrating gives y = x⁴ − x² + C. Since y(1) = 5, C = 5, so y = x⁴ − x² + 5.

How are position, velocity, and acceleration related?

v(t) = s′(t) and a(t) = v′(t) = s″(t). Thus, integrate velocity to obtain position and acceleration to obtain velocity.

Find s(t) when v(t) = 6t − 4 and s(0) = 3.

Integrate v(t): s(t) = 3t² − 4t + C. Using s(0) = 3 gives C = 3, so s(t) = 3t² − 4t + 3.

What idea does u-substitution reverse?

Substitution is integration by reversing the chain rule. Set u = g(x), replace g′(x) dx with du, integrate in u, and substitute back.

Evaluate ∫ 6x(3x² + 4)⁴ dx.

Let u = 3x² + 4, so du = 6x dx. Then ∫u⁴ du = u⁵/5 + C, giving (3x² + 4)⁵/5 + C.