Calculus is one of the important branches of mathematics. It includes concepts like integration and differentiation. Integration can be defined as the process of finding the antiderivatives of a given function. The anti-derivatives function is also known as the integral of a function. On the other hand, differentiation is the process by which we determine the derivatives of a function. Integration calculus is considered the inverse of differentiation. Derivatives can be defined as the rate of change of one quantity with respect to the other quantity. For example, speed. It is the rate of change in direction with respect to the time given.
Likewise, velocity is the rate of change in displacement with respect to the time given. Hence, the main application of integration calculusis to determine the anti-derivatives of a function. You must note that the inverse method or process of determining derivatives helps in finding the integrals. These integrals express or represent the family of the curve integrals expresses or represent the family of the curve. The finding of integrals and derivatives form the fundamentals of calculus.
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Types of Integrals
To recall, integration is the process of finding the antiderivatives of a given function. Likewise, integrals express or represent the family of curves. The inverse method or process of determining derivatives helps in finding the integrals.
The integrals that do not have the pre-existing value of limits used to find the antiderivatives are known as the indefinite integrals. Here, the value of the integrals is indefinite which signifies that they are constant, C.
The integrals that consist of the pre-existing value of limits used to find the anti-derivatives are known as the indefinite integrals. Here, the value of the integrals is definite.
Differentiation Vs Integration
Both the concepts of integration and differentiation are branches of calculus, but they consist of some major differences. The following points mentioned below analyze those differences.
- Integration can be defined as the process of finding the antiderivatives of a given function whereas, differentiation is the process by which we determine the derivatives of a function.
- Differentiation is used to find the slope of a function whereas integration is used to find the area under the region of the curve.
Integration Formulas
The formulas that can be applied in the integration of trigonometric ratios, algebraic expression, inverse trigonometric functions, exponential functions, and logarithmic functions can be defined as the integration formulas. Integration is the process of uniting part of a whole. An algebraic expression is the combination of various terms such as multiplication, subtraction, addition, and division. It is also known as the variable expression. Trigonometric ratios can be defined as the ratios of the sides of the triangle. Some of them are sine, cosine, tan, and so on. The integration of the functions results in determining the original function for which the derivatives are given. The integration formulas are used to find the anti-derivatives of the function.
Some Integration Formulas
As mentioned above, the integration formulas are used to find the anti-derivatives of the function. These formulas are applied in the integration of trigonometric ratios, algebraic expressions, inverse trigonometric functions, and other functions. Some examples of the integration formulas are:
- The formulas for the integration of the trigonometric functions are, ∫cosx.dx = sinx + C
∫ sinx. dx = -cost + C. There are other formulas as well, but these are the basic ones.
- Some basic integration formulas:
. ∫ 1.dx = x + C
. 1/x.dx = log|x| + C
. ∫ tanx.dx =log|secx| + C
. ∫ cotx.dx = log|sinx| + C
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