Integraalilausekkeiden

Integraalilausekkeiden, often translated as integral expressions or integral calculus, forms a fundamental branch of mathematics concerned with the concept of integration. It is the inverse operation to differentiation. While differentiation breaks down a function into smaller parts (its rate of change), integration aims to reconstruct the original function from its rate of change or to find the area under a curve.

The most common interpretation of an integral is the area under the curve of a function. The definite integral of a function over an interval represents the net area between the function's graph and the x-axis within that interval. This concept has wide-ranging applications in physics, engineering, economics, and statistics, where it is used to calculate quantities like displacement from velocity, work done by a force, or probability.

The indefinite integral, also known as the antiderivative, is a family of functions whose derivative is the original function. It is denoted by an integral sign followed by the function and 'dx', representing the differential of the variable of integration. The fundamental theorem of calculus establishes a crucial link between differentiation and integration, stating that the definite integral of a function can be evaluated by finding its antiderivative and assessing it at the interval's endpoints.

There are various techniques for evaluating integrals, including substitution, integration by parts, and partial fraction decomposition. These methods are essential for solving complex integral expressions. The development of integral calculus is credited to Isaac Newton and Gottfried Wilhelm Leibniz in the 17th century, though roots of the concept can be found in ancient Greek mathematics.