An indefinite integral is the reverse process of differentiation. When we differentiate a function, we find its rate of change. Integration does the opposite - it finds the original function from its rate of change. For example, if we know that the derivative of a function is 2x, then the original function could be x squared.
The notation for indefinite integrals uses the integral sign, which looks like an elongated S. This is followed by the function we want to integrate, called the integrand, and then dx, which indicates we are integrating with respect to x. The result is the antiderivative F of x plus a constant C, called the constant of integration.
The constant of integration C is crucial because the derivative of any constant is zero. This means if F of x is an antiderivative of f of x, then F of x plus any constant C is also an antiderivative. The constant represents an infinite family of parallel curves, all with the same derivative but shifted vertically.
Here are some fundamental integration formulas. 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. For the natural logarithm, the integral of 1 over x is the natural log of absolute value x plus C. The exponential function integrates to itself. For trigonometric functions, the integral of sine x is negative cosine x plus C, and the integral of cosine x is sine x plus C. Let's see an example: the integral of 3x squared equals 3 times the integral of x squared, which gives us x cubed plus C.
Indefinite integrals have many practical applications. In physics, we use them to find position from velocity, or velocity from acceleration. They're essential for calculating areas under curves and solving differential equations. In engineering, they help solve problems involving rates of change. Remember that integration is the reverse process of differentiation, and you must always include the constant of integration C to represent the complete family of antiderivatives.