Definite vs Indefinite Integrals
Calculus is an important branch of mathematics, and differentiation plays a critical role in calculus. The inverse process of the differentiation is known as integration, and the inverse is known as the integral, or simply put, the inverse of differentiation gives an integral. Based on the results they produce the integrals are divided into two classes; definite and indefinite integrals.
More about Indefinite Integrals
Indefinite integral is more of a general form of integration, and it can be interpreted as the anti-derivative of the considered function. Suppose differentiation of F gives f, and the integration of f gives the integral. It is often written as F(x)=∫ƒ(x)dx or F=∫ƒ dx where both F and ƒ are functions of x, and F is differentiable. In the above form, it is called a Reimann integral and the resulting function accompanies an arbitrary constant. An indefinite integral often produces a family of functions; therefore, the integral is indefinite.
Integrals and integration process are at the core of solving differential equations. However, unlike the differentiation, integration does not follow a clear and standard routine always; sometimes, the solution cannot be expressed explicitly in terms of elementary function. In that case, the analytic solution is often given in the form of an indefinite integral.
More about Definite Integrals
Definite integrals are the much valued counterparts of indefinite integrals where the integration process actually produces a finite number. It can be graphically defined as the area bounded by the curve of the function ƒ within a given interval. Whenever the integration is performed within a given interval of the independent variable, the integration produces a definite value which is often written as a∫bƒ(x)dx or a∫b ƒdx.
The indefinite integrals and definite integrals are interconnected through the first fundamental theorem of calculus, and that allows the definite integral to be calculated using the indefinite integrals. The theorem states a∫bƒ(x)dx = F(b)-F(a) where both F and ƒ are functions of x, and F is differentiable in the interval (a,b). Considering the interval, a and b are known as the lower limit and the upper limit respectively.
Rather than stopping with real functions only, the integration can be extended to complex functions and those integrals are called contour integrals, where ƒ is a function of the complex variable.
What is the difference between Definite and Indefinite Integrals?
Indefinite integrals represent the anti-derivate of a function, and often, a family of functions rather than a definite solution. In definite integrals, the integration gives a finite number.
Indefinite integrals associate an arbitrary variable (hence the family of functions) and definite integrals do not have an arbitrary constant, but an upper limit and a lower limit of integration.
Indefinite integral usually gives a general solution to the differential equation.