Integrals, Exponential Functions, and Logarithms

Integral Calculus helps you find the indefinite integral of a function. These indefinite integrals are also called function integrals. The process of finding an indefinite integral of a function is called an integral. The reverse process of finding the derivative is to find the integral. The integral of a function represents a series of curves. Finding the derivative and the integral form the basic calculus. This topic describes the basics of integration and the evaluation of integration.

Integral

The integral is the value of the function found in the integration process. Integration is the process of obtaining f (x) from f’ (x). Integral assigns numbers to functions in a way that describes displacement and motion problems, area and volume problems, etc. that result from combining all the small data. You can use the derivative f` of the function f to determine the function f. The function f is here called the indefinite integral or integral of f`.

F (x) is called the indefinite integral or Newton-Leibniz integral or primitive of the function f (x) of the interval I. F` (x) = f (x), for any value of x in I. 

 Integral is an expression. Of the area of the area below the curve. Approximate the true value of the integral by drawing a rectangle. The definite integral of a function can be expressed as the area of the area enclosed by the graph of a particular function between the two points of the line. The area of the area is calculated by dividing it into thin vertical rectangles and applying the lower and upper bounds, and the area of the area is summed. Specifies the integral of the function in the interval where the integral is defined.

Types of Integrals

 Integral calculations are used to solve the following types of problems: 

 a) The problem of finding a given function with a derivative. 

 b) The problem of finding the area enclosed by the graph of a function under given conditions. Therefore, integral calculations can be divided into two types. 

1. Definite integral (integral value is unique) 

2. Indefinite integral (the value of the integral is constant C and is indefinite)

Indefinite Integral

Given the function f (x), the indefinite integral of f (x) is any function F (x) such that F'(x) = f (x). If F (x) is an indefinite integral of f (x): The most common indefinite integral of f (x) is called an indefinite integral, and ∫f (x) dx = F (x) + c, c indicates that it is an arbitrary constant. In this definition, ∫ is called the integral symbol, f (x) is the integrand, x is the integral variable, and “c” is the integral constant.

The process of finding an indefinite integral is called an integral or an integral of f (x). When we need to be more specific about the variables of the integral, we say that we integrate f (x) with respect to x.

Definite Integral

The definite integral is the area below the curve between the two fixed limits. The definite integral is expressed as ^b∫_af(x)dx Here, for the function f (x) defined for the x-axis, a is the lower bound and b is the upper bound. Divide the area into rectangles and add them together to find the area below the curve between the two boundaries. The area is more accurate the more rectangles there are. Therefore, divide the area into an infinite number of rectangles, each of the same (very small) size, and sum all the areas. This is definitely the basic theory behind

Integrals of Exponential Functions

The format of the exponential function is f (x) = Ce ^{x} f (x) = Ce^x for the constant C and the linear shift, reciprocal, and quotient of such functions. Exponential functions are common in science, so it would be very useful to be able to integrate them. 

The exponential function is probably the most efficient function from an arithmetic point of view. The exponential function y = ex is its own derivative and integral.

The following formula can be used to combine exponential functions:

∫e^x dx = e^x + C

∫a^xdx = a^x/lna + C 

CONCLUSION

The integrals listed here are called definite integrals and can be interpreted as signed regions of the area in the plane surrounded by the graph of the definite integral between two points on the real line. Traditionally, the area above the horizontal axis of the plane is positive and the area below is negative. Integral is also related to the concept of indefinite integral. The derivative of this function is a given function. In this case, they are called indefinite integrals. The Fundamental Theorem of Calculus provides a way to combine definite and derivative to calculate the definite integral of a function when the indefinite integral is known.