Antiderivative Definition: every key term you need

Antiderivative Definition

F is antiderivative of f if F'(x) = f(x); also called indefinite integral; general form includes constant of integration.

Indefinite Integral Notation

∫f(x)dx = F(x) + C; represents family of antiderivatives; C is arbitrary constant.

Power Rule Integration

∫x^n dx = x^(n+1)/(n+1) + C (n ≠ -1); reverse of power rule for derivatives.

Integral of 1/x

∫(1/x)dx = ln|x| + C; absolute value needed because logarithm of negative requires careful handling.

Exponential Integration

∫e^x dx = e^x + C; ∫a^x dx = a^x/ln(a) + C; exponential functions integrate to themselves (up to constant).

Trigonometric Integration

∫sin(x)dx = -cos(x)+C, ∫cos(x)dx = sin(x)+C, ∫sec²(x)dx = tan(x)+C; reverse of trig derivatives.

Linearity of Integration

∫[af(x)+bg(x)]dx = a∫f(x)dx + b∫g(x)dx; integral of sum is sum of integrals; constant factors pull out.

U-Substitution

Let u = g(x), du = g'(x)dx; ∫f(g(x))g'(x)dx = ∫f(u)du; reverses chain rule; fundamental technique.

Integration by Parts

∫u dv = uv - ∫v du; useful for products; choose u via LIATE (Logarithm, Inverse trig, Algebraic, Trig, Exponential).

Partial Fractions

Decompose rational function into simpler fractions; integrate each piece; useful for rational functions with factorable denominators.

Trig Substitution

√(a²-x²): use x=a·sin(θ); √(a²+x²): use x=a·tan(θ); √(x²-a²): use x=a·sec(θ); converts to trig integrals.

Riemann Sum

Approximation of integral using rectangles; Σf(x_i*)Δx; left, right, midpoint rules; approaches integral as Δx→0.

Definite Integral Definition

∫_a^b f(x)dx = lim(n→∞) Σf(x_i*)Δx; area between curve and x-axis from a to b.

Fundamental Theorem of Calculus Part 1

If F'(x) = f(x), then ∫_a^b f(x)dx = F(b) - F(a); connects antiderivatives to definite integrals.

Fundamental Theorem Part 2

d/dx[∫_a^x f(t)dt] = f(x); derivative of integral reverses integration (and vice versa).

Area Between Curves

∫_a^b [f(x)-g(x)]dx where f(x)≥g(x); compute area of region between two curves.

Volume of Solids of Revolution

Disk method: V=π∫_a^b [R(x)]² dx; shell method: V=2π∫_a^b x·f(x)dx; rotate region around axis.

Average Value of Function

f_avg = (1/(b-a))∫_a^b f(x)dx; mean value; height of rectangle with same area as region under curve.

Improper Integrals

∫_a^∞ f(x)dx = lim(b→∞)∫_a^b f(x)dx; integrals with infinite limits or discontinuities; may diverge or converge.

Convergence of Improper Integrals

Test convergence by comparing to known functions (comparison test) or using limits; diverges if lim≠finite.

More AP Calculus AB guides

• Limits and Continuity
• Derivative Definition
• Extreme Value Theorem
• Differential Equations Basics