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

More AP Calculus BC guides

• Limits & Continuity
• Derivatives
• Applications of Derivatives
• Applications of Integrals
• Parametric & Polar

Antiderivative

F'(x) = f(x). Reverse of derivative. ∫f(x)dx = F(x) + C. Indefinite integral; +C accounts for constant.

Power Rule (Integration)

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

Exponential Integration

∫e^x dx = e^x + C. ∫a^x dx = a^x / ln(a) + C.

Logarithmic Integration

∫(1/x)dx = ln|x| + C. Absolute value accounts for negative x.

Trigonometric Integration

∫sin(x)dx = -cos(x) + C. ∫cos(x)dx = sin(x) + C. ∫sec²(x)dx = tan(x) + C.

Definite Integral

∫[a,b] f(x)dx = F(b) - F(a). Fundamental Theorem of Calculus. Represents signed area under curve.

Area Under Curve

∫[a,b] f(x)dx if f>0. If f<0, integral negative; use |∫| for area.

Riemann Sums

Approximate integral using rectangles. Left, right, midpoint, or trapezoidal. Converges to definite integral as n→∞.

Substitution (u-substitution)

Let u = g(x), du = g'(x)dx. Transform integral: ∫f(g(x))g'(x)dx = ∫f(u)du. Reverse of chain rule.

Integration by Parts

∫u dv = uv - ∫v du. Choose u, dv to simplify. Remember LIATE: Log, Inverse trig, Algebraic, Trig, Exponential.

Unit 4 Summary

Antiderivatives reverse derivatives. Definite integrals represent area. Substitution and by-parts handle complex integrands.

Riemann Sum Types

Left, right, midpoint, and trapezoidal sums approximate area. For an increasing function, left sums underestimate and right sums overestimate.

Trapezoidal Error Direction

Trapezoidal sums overestimate on concave-up intervals and underestimate on concave-down intervals; midpoint sums err in the opposite direction.

Definite Integral as a Limit

∫(a to b) f dx = lim(n→∞) Σ f(x_i*)Δx. Recognizing a limit of sums as an integral converts an impossible limit into a routine antiderivative.

Fundamental Theorem Part 1

If g(x) = ∫(a to x) f(t)dt with f continuous, then g'(x)=f(x). The integral is an antiderivative built by accumulation.

FTC with Variable Bounds

d/dx ∫(a to u(x)) f(t)dt = f(u(x))·u'(x). With both bounds variable, subtract the lower-bound contribution.

Fundamental Theorem Part 2

∫(a to b) f dx = F(b) - F(a) for any antiderivative F. This is what turns area into algebra.

u-Substitution

Choose u so that du appears (up to a constant) in the integrand; with definite integrals, convert the limits to u-values rather than back-substituting.

Integration by Parts

∫u dv = uv - ∫v du. Choose u by LIATE: logs, inverse trig, algebraic, trig, exponential — earlier types make better u.

Partial Fractions

Decompose a proper rational function with distinct linear factors into A/(x-r) + B/(x-s), then integrate each as a logarithm.

Improper Integral Convergence

Evaluate ∫(a to ∞) as lim(b→∞) ∫(a to b). ∫(1 to ∞) x^(-p) dx converges exactly when p > 1.

Discontinuous Integrand

An integral over an interval containing a vertical asymptote must be split at the asymptote and each piece taken as a separate limit.

Average Value of a Function

f_avg = (1/(b-a))∫(a to b) f dx. The Mean Value Theorem for integrals says a continuous f attains this value somewhere on the interval.

Even and Odd Symmetry

For symmetric limits, ∫(-a to a) of an odd function is 0, and of an even function is twice the half-interval integral.

Accumulation Function Analysis

Given a graph of f, the function g(x)=∫(0 to x) f dt increases where f>0, has extrema where f crosses zero, and inflects where f has extrema.

Net Change Theorem

∫(a to b) f'(x) dx = f(b) - f(a): integrating a rate returns the total change in the quantity over the interval.

Trig Antiderivative Set

∫tan x dx = -ln|cos x| + C, ∫sec x dx = ln|sec x + tan x| + C, and ∫sec²x dx = tan x + C.

Completing the Square

Rewrite quadratics under a radical or in a denominator as (x+h)² ± k² to reach an arctangent or arcsine form.

Long Division First

An improper rational integrand must be divided until the numerator degree is below the denominator degree before partial fractions apply.

Repeated Integration by Parts

For polynomial times exponential or sine, apply parts repeatedly, or use tabular integration to organize the alternating signs.

Comparison Test for Integrals

If 0 ≤ f ≤ g and ∫g converges then ∫f converges; if ∫f diverges so does ∫g. Useful when no antiderivative exists.