Suppose your friend gives you a wooden stick. He asks you to break it. Can you do so? Yes, it will be very easy for you to do so. But what will happen if he gives you five to six sticks to break? It will not be that easy to break it. As the number of sticks increases it is difficult to break them. The process of uniting things is an integration of things. Similarly, in mathematics too, we have an integration of two functions. Integration is like drop by drop addition of water in a container. Let us get ourselves familiar with the concepts of integrations.
In differentiation, we studied that if a function f is differentiable in an interval say, I, then we get a set of a family of values of the functions in that interval. Is there any way by which we can get to know about the function if the values of the function within an interval are known?
This process is the reverse of finding a derivative. Integrations are the anti-derivatives. Integrations are the way of adding the parts to find the whole. Integration is the whole pizza and the slices are the differentiable functions which can be integrated. If f(x) is any function and f′(x) is its derivatives. The integration of f′(x) with respect to dx is given as
∫ f′(x) dx = f(x) + C
Browse more Topics under Integrals
- Fundamental Theorem of Calculus
- Properties of Indefinite Integrals
- Properties of Definite Integrals
- Definite Integral as a Limit of a Sum
- Integration by Partial Fractions
- Integration by Parts
- Integration by Substitutions
- Integral of Some Particular Functions
- Integral of the Type e^x[f(x) + f'(x)]dx
The symbol for integration is S-shaped. Let us get familiar with some of the associated notations.
|∫ f(x) dx||Integral of f with respect to x|
|f(x) in ∫ f(x) dx||Integrand|
|x in ∫ f(x) dx||Variable of integration|
|dx in ∫ f(x) dx||Differentiation goes in the x direction|
|C||Constant of Integration|
Types of Integrations
There are two forms of the integrals.
- Indefinite Integrals: It is an integral of a function when there is no limit for integration. It contains an arbitrary constant.
- Definite Integrals: An integral of a function with limits of integration. There are two values as the limits for the interval of integration. One is the lower limit and the other is the upper limit. It does not contain any constant of integration.
Constant of Integration
The constant of integration expresses a sense of ambiguity. For a given derivative there can exist many integrands which may differ by a set of real numbers. This set of real numbers is represented by the constant, C.
Integration as an Inverse Process of Differentiation
Integration is the inverse process of differentiation. In integration, we have the derivative of a function and we need to find the original function.
|Derivatives||Integrals (Anti Derivatives)|
|d⁄dx (x n + 1 ⁄ n + 1) = xn||∫ xn dx = x n + 1 ⁄ (n + 1) + C, n ≠ −1|
|d ⁄ dx (x) = 1||∫dx = x + C|
|d⁄dx (sin x) = cos x||∫ cos x dx = sin x + C|
|d ⁄ dx (− cos x) = sin x||∫sinx dx = − cos x + C|
|d⁄dx (tan x) = sec 2 x||∫ sec 2 x dx = tan x + C|
|d ⁄ dx (− cot x) = cosec 2 x||∫ cosec 2 x dx = − cot x+ C|
|d ⁄ dx (sec x) = sec x tan x||∫(sec x + tan x)dx = sec x + C|
|d⁄dx (− cosec x) = cosec x cot x||∫(cosec x cot x)dx = − cosec x + C|
|d ⁄ dx (sin−1 x) = 1⁄ √(1 − x2)||∫ dx ⁄ √(1 − x2) = sin−1 x+ C|
|d⁄dx (− cos−1 x) = 1⁄ √(1 − x2)||∫dx ⁄ √(1 − x2) = − cos−1 x + C|
|d ⁄ dx (tan−1 x) = 1⁄ (1 + x2)||∫ dx ⁄ (1 + x2) =tan−1 x + C|
|d⁄dx (− cot−1 x) = 1⁄ (1 + x2)||∫dx ⁄ (1 + x2) = − cot−1 x + C|
|d ⁄ dx (sec−1 x) = 1⁄ x√(x2 − 1)||∫ dx ⁄ x√(x2 − 1) = sec−1 x + C|
|d⁄dx (− cosec−1 x) = 1⁄ x√(x2 − 1)||∫dx ⁄ x√(x2 − 1) = − cosec−1 x + C|
|d ⁄ dx (ex) = ex||∫ ex dx = ex + C|
|d⁄dx (log |x|) = 1⁄x||∫1⁄x dx = log |x| + C|
|d ⁄ dx (ax⁄ log a) = ax||∫ ax dx = ax⁄ log a + C|
Properties of Indefinite Integrals
The process of differentiation and integration are inverses of each other.
Proof: Let F be an anti-derivative of f, i.e., d⁄dx F(x) = f(x)
Then ∫ f(x) dx = F(x) + C
Therefore, d⁄dx ∫ f(x) dx = d⁄dx [F(x) + C] = d ⁄ dx F(x) = f(x). Similarly,
f′(x) = d⁄dx f(x) and hence ∫ f′(x) dx = f(x) + C. C is the constant of integration.
The integration of the sum of two integrands is the sum of integrations of two integrands.
∫ [f(x) + g(x)] dx = ∫ f(x) dx + ∫ g(x) dx
Proof: Using theorem 1, we have
d⁄dx [∫ [f(x) + g(x)] dx] = f(x) + g(x) … (1)
Also, d⁄dx [∫ f(x) dx + ∫ g(x) dx] = d⁄dx ∫ f(x) dx + d⁄dx ∫ g(x) dx = f(x) + g(x) … (2)
From (1) and (2), we have, ∫ [f(x) + g(x)] dx = ∫ f(x) dx + ∫ g(x) dx
For any real number k, ∫ k f(x) dx = k ∫ f(x) dx.
Proof: Using theorem 1, d⁄dx ∫ k f(x) dx = k f(x) … (1)
Also, d⁄dx [∫ k f(x) dx] = k d⁄dx ∫ f(x) dx = k f(x) … (2)
From (1) and (2), we have, ∫ k f(x) dx = k ∫ f(x) dx.
The above result can be generalised to ʃ k1 [f1(x) + k2 f2(x) + … + kn fn(x)] dx = k1 ∫ f1(x) dx + k2 ∫ f2 (x) dx + … + kn ∫ fn(x) dx.
Solved Examples for You
Example 1: Solve ∫ (x5 – 1) ⁄ x2 dx.
Solution: ∫(x5 – 1) ⁄ x2 dx = ∫x3 dx – ∫x–2 dx = x 3+1 ⁄ (3 + 1) + C1 – (x–2+1 ⁄ (–2 + 1)) + C2
or, ∫ (x5 – 1) ⁄ x2 dx = x4 ⁄ 4 + 1⁄x + C1 + C2 = x4 ⁄ 4 + 1⁄x + C where, C = C1 + C2
Example 2: Solve ∫ (cos x – sin x) dx.
Solution: ∫ (cos x – sin x) dx = ∫ cos x dx – ∫ sinx dx = sin x – (– cos x) + C = sin x + cos x + C.
Ques. What is the integration used for?
Ans. From a very common and basic point of view, integration is used for measuring things. It might be a length, area, or volume. It can also be probabilities in the context of unplanned variables. It helps in measuring the behavior of a function. In any specific sense that one wants to measure the behavior.
Ques. What is an example of integration?
Ans. Integration is also stated as the process of mixing things or people together. It mixes the things that were separated. A common example of integration is that time when the schools were merged and there were no longer separate schools for the African Americans.
Ques. Why is system Integration Important?
Ans. The system integration is becoming much essential due to the increasing advancement in the automation technology sector. Also due to the associated necessity to simplify the procedures for easier controlling and management. An integrated method will update the processes, reduce the costs and safeguard the efficiency.
Ques. What are the advantages of integration?
Ans. Horizontal integration can benefit the companies and normally takes place when they are challenging in the same industry or sector. These advantages comprise of the growing market share, decreasing competition, and making economies of scale.