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Introduction to Integration

Updated: Jun 22, 2021



In this note, you will learn:


· What is integration?


· Rules of integration:


1. Scalar Multiple Rule

2. Addition/Subtraction Rule of Integration



· Integration of power functions of the form (ax + b)^n


What is integration?

Integration, in mathematical terms, is the assignation of numbers to functions in a manner to describe displacement, area, volume and many other relevant concepts that arise from the combination of infinitesimal (numbers that are closer to zero compared to any standard real number, but does not equate to zero) data.



Integration is one of the two main operations of calculus, the other being differentiation, which is the inverse operation of integration.



The symbol used in integration is ‘∫’, called the integration sign. The function that is being integrated is called an integrand.



The general rule of integration to find the integral of a function is to actually very simple, just imagine it to be the complete reverse operation of a differentiation. What we mean by this is for differentiation, we:



- Multiply the term you are trying to differentiate with the initial power of the variable


- Decreasing the power of the variable by 1


- Remove any constant terms as the derivative of a constant is zero



However, for integration, we flip the order of the steps and:


- Increase the power of the variable by 1


- Divide the term by the value of the increased power


- Add a constant term, c, at the end of the integration



The reason why we add a constant term at the end of our answers when dealing with indefinite integrals is because during the process of differentiation, constant terms become zero. Hence, when we integrate, we must compensate for any constant term.



Therefore, the general formula for integration is given by:


∫ x^n dx = (x^(n+1))/ (n+1),

where n ≠ -1 and c is a constant



Similar to differentiation, for integration once we write down the integral sign and the function we are trying to integrate, we must add a ‘dx’ at the back to indicate that we are integrating the function with respect to x.



Let’s take a look at some simple examples below to help you understand the concept of indefinite integration:



Find the indefinite integral of each of the following.


a) x^4

b) dy/dx = 1

Solution:

a) ∫ x^4 dx = [(x^(4+1))/ (4 + 1)] + c

= (x^5/5) + c


b) dy/dx = 1

y = ∫ 1 dx

= ∫ x^0 dx

= [x^(0+1)/ (0 + 1)] + c

= x + c


Rules of integration


· Scalar Multiple Rule


The Scalar Multiple Rule suggest that considering the function k f(x), where k is a constant and f(x) is a function, k will hence be called a scalar multiple. Just like for differentiation, this means that k, the scalar multiple, can be moved to the front of the term and be omitted from the integration of the term.



The scalar multiple can then be multiplied back into the result once the integration of the term is done. This is as shown below:


∫ k f(x) dx = k ∫ f(x) dx


· Addition/Subtraction Rule of Integration


The Addition/Subtraction Rule suggests that if f(x) and g(x) are functions, then the integration when there is an addition or subtraction of the two functions can be split into the integration of two individual functions. This is as shown below:


∫ [f(x) +/- g(x)] dx = ∫ f(x) dx +/- ∫ g(x) dx


Let’s take a look at how we can apply these rules of integration in some example questions down below:


1. Integrate each of the following with respect to x.


a) -4/9x^5


b) dy/dx = k



Solution:


a) ∫ (-4/9x^5) dx = ∫ (-4/9) x^-5 dx

= (-4/9) ∫ x^-5 dx (Rule 1 of Integration)

= (-4/9) (x^-4/-4) + c

= (1/9 x^4) + c



b) dy/dx = k

Therefore, y = ∫ k dx

= ∫ k x^0 dx

= k ∫ x^0 dx (Rule 1 of Integration)

= k (x^1 /1) + c

= kx + c


2. The gradient of a curve is given by dy/dx = (4x – 3) (2x + 5). Given that the curve passes through the point (2, 3) find the equation of the curve.

Solution:

dy/dx = (4x – 3) (2x + 5)

= 8x^2 + 20x – 6x -15

= 8x^2 + 14x – 15


Therefore, y = ∫ (8x^2 + 14x – 15) dx

= (8x^3/3) + (7x^2) – (15x) + c (Rule 2 of Integration)



Since the curve passes through (3, 5), then 5 = [8(3)^3 /3] + [7(3)^2] – [15(3)] + c


Hence, c = 5 – 72 – 63 + 45

= -85


Therefore, the equation of the curve is y = (8/3) x^3 + 7x^2 – 15x – 85.




Integration of power functions of the form (ax + b)^n


In the chapter of differentiation, we have learnt how to differentiate power functions of linear factors using the Chain Rule. For example,

d/dx [(5x + 3)^6] = 6 (5) (5x + 3)^5


If we were to consider the concept that differentiation is a reverse operation of integration, we will get:

∫ 6 (5) (5x + 3)^5 dx = (5x + 3)^6 + c

6 (5) ∫ (5x + 3)^5 dx = (5x + 3)^6 + c


If we were to divide both sides by 6 (5), we will get:

∫ (5x + 3)^5dx = [(5x + 3)^6]/ [6(5)] + c1, where c1 = c/ [6(5)]


Since the constant of integration is arbitrary, we can write

∫ (5x + 3)^5 dx = [(5x + 3)^6/ [6(5)]


In conclusion, we will end up with the following general formula:

∫ (ax + b)^n dx = [(ax + b)^n/ (n+1) (a)] + c,

where n ≠ -1 and a ≠ 0


Let’s take a look at how we can apply this integration formula in the examples down below:



Integrate each of the following with respect to x.

a) (3x + 8)^9

b) 5/ √(4x + 7)


Solution:

a) ∫ (3x + 8)^9 dx = [(3x + 8)^9+1 / (9+1) (3)] + c

= (1/30) (3x + 8)^10 + c

b) ∫ (5/ √4x + 7) dx = 5 ∫ (4x + 7)^-1/2 dx

= (5) {(4x + 7)^(-1/2] +1/ [(-1/2) + 1] (4)} + c

= 5 [(4x + 7)^ (1/2)/ (4/2)] + c

= (5/2) √ (4x + 7) + c


And that’s all for today, students! Math Lobby hopes that after this article, you have a clear understanding on the introduction to the topic of integration, and is equipped with the necessary skills to deal with questions involving the application of the different rules of integration!



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