Integral Rules of Algebraic functions -Formulas

In integral calculus, there are three types of algebraic integral rules. They are employed as formulas in the calculation of both indefinite and definite integrals of algebraic functions.

Integration is the process of computing an integral. In mathematics, integrals are used to calculate numerous useful quantities, including areas, volumes, displacements, etc. Typically, when we discuss integrals, we refer to definite integrals. Antiderivatives are derived using indefinite integrals.

Power Rule-

In calculus, functions of the form displ4are differentiated using the power rule whenever display style r is a real number. Since differentiation is a linear operation on the space of differentiable functions, this rule can also be applied to the differentiation of polynomials. Taylor series is based on the power rule, which relates a power series to a function’s derivatives.

The power rule for integrals was first demonstrated in a geometric form by the Italian mathematician Bonaventura Cavalieri in the early 17th century for all positive integer values n, and then independently by Pierre de Fermat, Evangelista Torricelli, Gilles de Roberval, John Wallis, and Blaise Pascal in the middle of the 17th century for all rational powers. They were treatises on calculating the area between the graph of a rational power function and the horizontal axis at the time. Nonetheless, it is considered the first general theorem of calculus discovered in retrospect. Isaac Newton and Gottfried Wilhelm Leibniz independently derived the power rule for differentiation for rational power functions in the middle of the 17th century, and then used integration in order to derive the power rule for integrals as the inverse operation. This is consistent with how related theorems are typically presented in contemporary textbooks for basic calculus, where differentiation rules typically precede integration rules.

Exponential integral rules-

Exponential functions are utilised to model population growth, cell growth, and financial growth, among other applications, as well as depreciation, radioactive decay, and resource consumption.

The exponential function is possibly the most effective function in terms of calculus operations. The exponential function, y=ex, is the derivative and integral of itself. Exponential functions are those whose independent variable, x, is the power or exponent of the base. 

Rule: Exponential function integrals 

The definition of an elementary exponential function is f(x) = bx, where b is a constant and x is a variable. Popular exponential functions include f(x) = ex, where e is “Euler’s number” and e = 2.718…. Extending the possibilities of various exponential functions, a function may include a constant as the multiple of the variable within its power. In other words, an exponential function may also take the form f(x) = ekx. In addition, it may take the form f(x) = p ekx, where p is a constant. Consequently, an exponential function can take the following forms:

1. f(x) = bx

2. f(x) = abx

3. f(x) = abcx

4. f(x) = ex

5. f(x) = ekx

6. f(x) = p ekx

Here, all letters except ‘x’ are constants, ‘x’ is a variable, and f(x) is an exponential function with respect to x. Note that the base of every exponential function must be positive. Therefore, in the preceding functions, b > 0 and e > 0 Also, b should not equal 1 (if b = 1, the function f(x) = bx becomes f(x) = 1, in which case the function is linear and NOT exponential).

Reciprocal Integral Rules-

The reciprocal rule in calculus relates the derivative of the reciprocal of a function f to the derivative of f. The reciprocal rule can be used to demonstrate that the power rule applies to negative exponents if its validity for positive exponents has already been established. Additionally, the quotient rule can be easily deduced from the reciprocal rule and the product rule.

If f is differentiable at a point x and f(x) 0, then g(x) = 1/f(x) is also differentiable at x and the reciprocal rule holds.

Conclusion-

An integral in mathematics assigns numbers to functions in a way that describes displacement, area, volume, and other concepts resulting from the combination of infinitesimal data. The procedure of determining integrals is known as integration.

Integrity is the opposite of differentiation. Therefore, the integral of 2 can be 2x plus 3, 2x plus 5, 2x, etc. Integrating, therefore, requires the addition of a constant. Therefore, the integral of 2 is 2x plus a constant.