Integration is a summation operation that may be used as a mathematical tool for determining the area with functions of a single variable, computing the surface area and volume of three-dimensional solids, calculating the area and volume of a function with two variables, or summing multidimensional functions.
Real-world quantities such as temperature, magnetic field strength, pressure, speed, flow rate, lighting, share prices, and so on can be defined mathematically in science, engineering, and economics. We can use integration to combine these variables to get a total result.
In this post, we will explain the rules of integration with comprehensive examples and their solutions. This post is intended for the students of calculus and it will help you to understand the concept of integrals along with the rules.
Integration Rules – The Need
When it comes to integrating functions or taking antiderivatives, the same basic rules apply as they do for differentiation. This antiderivative calculator uses all of the rules while calculating antiderivatives. You should be familiar with the notion that integration and differentiation are the opposites of one another.
We can always distinguish the result to go back to the original function if we integrate a function. However, this is not the case. Because the derivative of any constant term is zero, any constant term in a function usually disappears when it is differentiated.
It is something we should bear in mind while considering how to integrate a function because it implies that our solution will always contain a constant with an unknown value. This constant is known as the constant of integration, C.
The most important rule of integration is the power rule of integration. This method is effectively the reverse of the power rule used in derivatives, and it yields the indefinite integral of a variable raised to a certain power. To refresh your memory below is the integration power rule formula:
The indefinite integral of the variable x raised to the power of n multiplied by the constant-coefficient a is given by this formula. Also bear in mind that n cannot be equal to -1, because, on the right-hand side of the formula, this would put a 0 in the denominator. This criterion alone allows us to integrate polynomial functions using a single variable.
We just integrate each expression independently, with no modification to the plus or minus sign in front of each word. Some typical indefinite integrals are listed below. It’s worth noting that in these instances, a stands for a constant, x stands for a variable, and e stands for Euler’s number which is approximately 2.7183. It’s also worth noting that the first three instances are the result of applying the power rule.
Indefinite Integrals
A constant value a:
∫ a dx = ax + C
A variable x:
The square of a variable x^{ 2}:
The reciprocal of a variable ^{ }^{1}/_{x}:
The exponential function e^{ x}:
Other exponential functions a^{ x}:
The natural logarithm of a variable ln (x):
The sine of a variable sin (x):
The cosine of a variable cos (x):
∫ cos (x) dx = sin (x) + C
The power, constant-coefficient or constant multiplier, sum, and difference rules are among the basic integration rules that will be discussed here. We’ll give some easy examples to show how these integration principles and laws actually work. Before moving onto the next section, check out this Integral calculator with steps. It can help you to find the indefinite integral of any given function.
The Power Rule
As we have seen, the power rule for integration is the inverse of the power rule for differentiation. It returns the indefinite integral of a variable multiplied by a power. Here’s the power rule again:
Let’s look at some examples of how this rule is used. Assume we wish to calculate the indefinite integral of x^{3}. Using the power rule:
It is not always evident that we can apply the power rule to get the indefinite integral of a function. Assume we wish to calculate the indefinite integral of the equation ^{3}√x. How can we apply the power rule to the cubed root function? It’s actually fairly simple. All we have to do is change the expression to get x to a power. To express the nth root of a number in an index form, there is a common formula which can be stated as:
^{n}√a = a^{ 1/}_{n}
Applying this formula to ^{3}√x:
^{3}√x = x^{ 1/}_{3}
We can now apply the power rule to get:
The Constant Coefficient Rule
The constant-coefficient rule is also known as the constant multiplier rule. It states that the indefinite integral of c∙f(x), where c represents a constant coefficient and f(x) is some function, is equivalent to the indefinite integral of f(x) multiplied by c. This can be stated as follows:
The constant-coefficient rule allows us to disregard the constant-coefficient in an equation while integrating the remainder of it. Let’s say we wish to compute the indefinite integral of the expression 3x^{2}. According to the constant-coefficient rule, the indefinite integral of this equation is the indefinite integral of x^{2} multiplied by 3. That is to say:
Now we just apply the power rule to x^{ 2}:
The Sum Rule
The sum rule describes how to integrate functions that are the sum of many terms. It simply shows us that we must integrate each expression independently in the total, before adding the results together. It is unimportant which order the terms appear in the outcome. This can be stated as follows:
You may be asking why the regulation is worded the way it is at this point. It is critical to understand that in a function that is the sum of two or more components, each term may be thought of as a function in its own right – even a constant term. Assume we wish to calculate the indefinite integral of a function (x) = 3x^{2} + 4x + 12. Using the sum rule:
The Difference Rule
The difference rule instructs us on how to integrate functions that include the difference of two or more terms. It is similar to the sum rule in that it instructs us to integrate each term in the sum independently. The only distinction is that the order of the expressions is important and cannot be modified. This rule can be stated explicitly as follows:
Let’s have a look at an example. Assume we wish to calculate the indefinite integral of the polynomial function (x) = 5x^{3} – 9x – 2. Using the sum rule, we obtain:
The difference and sum rules are fundamentally the same rules. If we wish to integrate a function that comprises both the sum and difference of a number of terms, we must remember to integrate each term independently and to keep the order of the terms in mind. The “+” or “=” sign in front of each expression remains the same.
You may also conceive the function as the sum of a number of positive and negative terms and apply the sum rule. The order is thus irrelevant; you just need to be aware of the sign of each expression. Below, we have listed few more examples for further interpretation of integration rules.
Example 1:
Evaluate ∫ 7 dx
∫ 7 dx =
7 ∫ dx ……….multiplication by a constant rule
= 7x + C
Example 2:
What is ∫ 5x^{4} dx
∫ 5x^{4} dx = 5 ∫x^{4} dx ……. using multiplication by a constant rule
= 5(x^{5}/5) + C ………. using power rule
= x^{5} + C
Example 3:
Evaluate ∫ (2x^{3} + cos(x) ) dx
∫ (2x^{3} + 6cos(x) ) dx = ∫ 2x^{3} dx + ∫ 6cos(x) dx …..Applying the sum rule
= 2 ∫ x^{3} dx + 6 ∫ cos(x) dx ……….Applying the multiplication by a constant rule
= 2(x^{4}/4) + C_{1 }+ 6(sin(x) + C_{2} …..Applying the power rule. C_{1} and C_{2} are constants.
C_{1} and C_{2 }can be replaced by a single constant C, so:
∫ (2x^{3} + cos(x) ) dx = x^{4}/2 + 6sin(x) + C
Summary
All of the listed rules are extensively used in integration and are vital for the evaluation of integrals. These rules should be practiced and implemented on several types of functions if you want to master the concept of integrals. Refer to these principles listed above if you are stuck somewhere while calculating antiderivative or integral.