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Integration and Differentiation 

 

The two main ideas around which calculus is made are differentiation and integration. Differentiation is used to investigate the tiny change of one quantity concerning the unit change of another. On the other hand, integration is used to integrate tiny and discontinuous data that cannot be added individually and represented in a single value. The rate of change of temperature is a good demonstration of differentiation, and the best example of integration is determining the region between the curves for bigger firms. Differentiation and integration are essential methods in calculus that are used to address math and science issues.

Integration

Integration is a method of connecting parts to form a whole. We find a function whose differential is provided in integral calculus. As a result, differentiation is the inverse of integration. The area of the region enclosed by the graph of functions is defined and calculated using integration. Tracking the integrand of the polygon engraved in the curved form approximates its area. Integration of a function is represented with this ‘ʃ’ symbol. We get two kinds of integrals: indefinite and definite.

Indefinite Integrals: As the name suggests, a function’s integral when there is no limit to integration is called indefinite integral. It includes an arbitrary constant represented by the symbol ‘C’.

Definite Integrals: A function integral having integration limitations is expressed as a definite integral. The interval of integration has two values as its boundaries. The lower value is the lower limit, while the higher value is the upper limit. There is no integration constant in it.

Famous mathematician Leibniz developed the concepts of integration. There are four basic methods of integration which are

  • Integration by Decomposition method

  • Integration by Substitution

  • Integration Using Partial Fractions

  • Integration by Parts

Differentiation

In mathematics, differentiation is a part of calculus defined as the derivation of any function using an independent variable. It can also be referred to as the function per unit change. It is denoted by dy/dx or y’, where x is the independent variable and y is differentiated concerning x. If any function has an extremely small change ‘k’ close to x, then the derivative is known as lim k→0 {f(x+k)–f(x) / k}

There are four types of differentiation rules:

Product Rule: d/dx (pq) = q * dp/dx + p * dq/dx

Sum and Difference Rule: F(X) () F(Y) = F’(X) () F’(Y)

Quotient Rule: d/dx {p(x) / q(x)} = [q(x) * p’(x) – p(x) * q’(x)]/ p(x)2

Chain Rule: The differentiation of a mixed function is determined by this rule. The idea is to solve the exterior function first, followed by the interior function. For instance, tan(k²) is a composite function because it can be expressed as m(n(x)) for m(x) = tan(k) and n(x)=k².

Properties of Differentiation and Integration

  • Differentiation and integration are both procedures that need limitations to be determined.
  • When a polynomial function is differentiated, the result has a degree that is one less than the degree of the polynomial function, but when a polynomial function is integrated, the result has a degree that is one greater than the degree of the polynomial function.
  • As previously stated, differentiation and integration are inverses of each other.
  • The derivative of each function is unique; nevertheless, the integral of any function is not unique. A constant can separate two integrals of the same function.
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