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Integration is an important part of calculus. Integrals include single integral, double integral, and multiple integrals. Various types of integral are used to find surface area and the volume of geometric solids.
The double integral, triple integral are some of the techniques. The mostly used ones are Gauss divergence theorem and Stokes theorem in vector calculus. The Gauss divergence theorem produces results which relates the flow of the vector field vector field through a surface to the behavior of the vector field within the surface.
Integration is a process of the summation of a product. In fact, the integration symbol $\int $ is actually an elongated S, the S meaning a summation.
Consider a function $f(x)$ when it undergoes an infinitesimal change of $dx$. The product of the function and the infinitesimal change at any point is $f(x)dx$. In other words, it is the area of an infinitely small rectangle of the height $f(x)$ and width $dx$. The summation of such areas in all points in the domain of the function is called the integration of the function and is denoted as $\int f(x)dx$, $\int f(x)dx$. The entire term is called as the integral of $f(x)$.
In another concept, integration is defined as the inverse process of differentiation and hence the evaluation of an integral is called as anti derivative. Having said about integration as the product described above, we have not said the summation is to be done from which point to which
point. In other words, the interval of summation is indefinite and hence these types of integrals are known as indefinite integrals.
The anti derivative of a definite integral is only implicit, that is, the solution will only be in a functional form. That is, $\int f(x) dx = g(x) + C$, where $g(x)$ is another function of $x$ and $C$ is an arbitrary constant. However, there are many integrals which are to be integrated within a given interval. They are denoted in general as $\int_^f\left ( x \right )dx$ where, a and b are the limits of the interval. Such types of integrals are known as definite integrals. The solution of a definite integral is unique and the solution to $\int_^f\left ( x \right )dx$ is $F(b) – F(a)$, where $F(x)$ is the anti derivative of the given integral.
Antiderivative of a Function
A function p(x) is called a primitive or an anti derivative of a function f(x) if p’(x) = f(x). Let p(x) be an anti derivative of a function f(x) and let C be any constant. Then,
Derivative of [p(x) + C] = p’(x) = f(x). This means p(x) + C is also an anti derivative of f(x).
Thus, if a function f(x) possesses an anti derivative, then it possess infinitely many anti derivatives which are contained in the expression p(x) + C, where C is a constant. For example, $\left ( \frac
Integration by Substitution