Each card is a single, complete idea: what it is, the formulas you need, and one worked example. Built to be re-read.
A definite integral measures the total change in a quantity over an interval. Geometrically, it is the signed area between the curve and the x-axis: regions above the axis are positive, and regions below are negative.
When a region is not made of simple shapes, approximate its area with rectangles or trapezoids. Left, right, and midpoint sums use one endpoint per sub-interval; the trapezoidal rule averages left and right. More sub-intervals give better accuracy.
Definite integrals behave like signed areas: equal bounds give zero, swapping bounds flips the sign, constants factor out, and integrals split over adjacent intervals.
An indefinite integral is the family of all antiderivatives. Always include +C because the derivative of any constant is zero. The reverse power rule raises the exponent by one and divides by the new exponent.
Integration reverses differentiation. Memorize the basic antiderivatives for exponentials, logarithms, and the six trigonometric functions. Watch the sign on sine and cosine.
Differentiation and integration are inverse operations. If an accumulation function has a variable upper limit, its derivative is the integrand evaluated at that limit. If the upper limit is a function of x, apply the chain rule by multiplying by the derivative of the bound.
To evaluate a definite integral, find any antiderivative F, then subtract its values at the bounds. This is the workhorse of integration on the AP exam.
u-Substitution reverses the chain rule. Choose u to be the inner function, compute du, and rewrite the integral so that the remaining dx terms match. Don't forget to change the bounds for definite integrals or substitute back for indefinite integrals.
When an integrand doesn't fit a basic rule, rewrite it algebraically. Split fractions, expand products, simplify radicals into powers, or use polynomial long division to make a 'top-heavy' rational function manageable. Completing the square can reveal inverse-trig forms (BC).