Integration is one of the fundamental operations in calculus. It's the reverse process of differentiation. While differentiation finds the rate of change, integration finds the area under curves. For example, if we have the function f of x equals x squared, its integral is x cubed over 3 plus a constant C. We can verify this relationship by taking the derivative of x cubed over 3, which gives us back x squared. This demonstrates the fundamental connection between integration and differentiation.
An antiderivative is a function whose derivative gives us the original function. If F prime of x equals f of x, then F of x is called an antiderivative of f of x. For example, the antiderivative of x squared is x cubed over 3 plus C. We can verify this by taking the derivative, which gives us x squared. Similarly, the antiderivative of 2x is x squared plus C. The constant C is crucial because the derivative of any constant is zero, so there are infinitely many antiderivatives that differ only by a constant.
Now let's learn the basic integration rules. The power rule states that the integral of x to the n is x to the n plus 1, divided by n plus 1, plus C. The constant rule says the integral of a constant k is k times x plus C. The sum rule allows us to integrate each term separately. Let's apply these rules to integrate 3x squared plus 2x plus 1. First, we separate the terms. Then we factor out constants. Applying the power rule: x squared becomes x cubed over 3, x becomes x squared over 2, and the constant 1 becomes x. Our final answer is x cubed plus x squared plus x plus C.
There are two types of integrals: indefinite and definite. An indefinite integral represents a family of functions and includes the constant C. It's written as the integral of f of x dx equals F of x plus C. A definite integral gives a specific numerical value and represents the area under a curve between two bounds. It's calculated using the Fundamental Theorem of Calculus: the integral from a to b of f of x dx equals F of b minus F of a. For example, the definite integral of x squared from 1 to 3 equals x cubed over 3 evaluated from 1 to 3, which gives us 27 over 3 minus 1 over 3, equals 26 over 3.
Let's work through three integration examples using our systematic approach. First, the integral of 2x plus 3. We apply the power rule to each term: 2x becomes x squared, and 3 becomes 3x, plus our constant C. Second, for x cubed minus 4x squared plus 5, we get x to the fourth over 4, minus 4x cubed over 3, plus 5x, plus C. Finally, for the definite integral from 0 to 2 of x squared plus 1, we first find the antiderivative: x cubed over 3 plus x. Then we evaluate at the bounds: 8 over 3 plus 2, minus 0, which equals 14 over 3. This represents the area under the curve, as shown in the graph.