Integration: every key term you need (+ practice quiz)

More AP Calculus AB guides

• Limits and Continuity
• Derivatives
• Applications of Derivatives
• Differential Equations

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.

Riemann Sum Bias

For an increasing function, a left sum underestimates and a right sum overestimates the definite integral; the inequalities reverse for a decreasing function.

Trapezoidal Estimate

Each subinterval contributes width times the average of its endpoint values; the sum overestimates for concave-up functions and underestimates for concave-down ones.

Definite Integral as Limit

∫ from a to b of f(x)dx = lim(n→∞) Σ f(x_i)Δx; the sum of signed areas of rectangles becomes exact area in the limit.

Second Fundamental Theorem

d/dx ∫ from a to x of f(t)dt = f(x); with an upper limit u(x) the chain rule adds a factor u'(x).

Accumulation Functions

For g(x) = ∫ from a to x of f(t)dt, the sign of f controls whether g increases, and the slope of f controls the concavity of g.

Substitution Rule

With u = g(x) and du = g'(x)dx, ∫ f(g(x))g'(x)dx becomes ∫ f(u)du; a definite integral should have its limits converted to u values.

Average Value

The average value of f on [a,b] is (1/(b-a))∫ from a to b of f(x)dx; the Mean Value Theorem for integrals says a continuous f attains it.

Integrals of Absolute Value

Split the interval at the point where the inside expression changes sign, integrate each piece, and add the resulting nonnegative areas.

Net Change Theorem

∫ from a to b of f'(x)dx = f(b) - f(a); integrating a rate over an interval gives the total change in the quantity.

Displacement vs Distance

Displacement is ∫ v(t)dt while total distance is ∫ |v(t)|dt; they differ whenever the velocity changes sign inside the interval.

Linear Inner Substitution

∫ (2x+1)⁵dx = (2x+1)⁶/12 + C; the extra 1/2 comes from the constant derivative of the inside function.

Integral of Tangent

∫ tan x dx = -ln|cos x| + C, found by substituting u = cos x so that du = -sin x dx.

Integral of 1/x

∫(1/x)dx = ln|x| + C; the absolute value is required so the antiderivative is valid on intervals where x is negative.

Basic Exponential Integral

∫e^(kx)dx = e^(kx)/k + C; dividing by k undoes the chain-rule factor that differentiation would introduce.

Secant-Squared Integral

∫sec²(kx)dx = tan(kx)/k + C, the direct reversal of the derivative of tangent with a linear inside function.

Properties of Definite Integrals

Reversing limits negates the value, an integral over a degenerate interval is 0, and integrals split additively at any interior point.

Even and Odd Symmetry

On [-a,a], an odd integrand integrates to 0 and an even integrand integrates to twice the value on [0,a].

Integrals from Geometry

When the integrand's graph is made of lines and circular arcs, evaluate by computing signed areas rather than antidifferentiating.

Units in Applied Integrals

The units of a definite integral are the integrand's units multiplied by the variable's units, so a rate in liters per minute integrates to liters.

Choosing u

Pick u so that its derivative appears (up to a constant) elsewhere in the integrand; if nothing cancels, the substitution choice was wrong.