It’s the branch of mathematics that deals with integrals and their characteristics. Moreover, it provides interesting information about the anti-derivatives of functions. It also provides the information determination of integral, properties, and their applications. Integration is a concept that finds the area under the curve.
For discussion of integral we have detailed knowledge of derivative or differentiation. In this article, we will thoroughly discuss integral calculus, along with its definition, types, formulas, rules, and examples.
Definition of Integral
Integral calculus is the continuous analog of a sum, which is used to calculate, Areas, volumes, and their generalizations.” It is represented by “∫”. If we know the ‘g’ of the functions that are differentiable in their domains, then we can calculate g. In Differentiate calculus g’ is recalling the derivative of the function g.
On the other hand, ‘g’ is related to the anti-derivative of ‘g’ in integral calculus. Anti-differentiation or integration are terms used to describe the process of identifying the anti-derivatives. It is the inverse process of derivatives.
Types of Integral
Generally, there are two basic types of the integral.
- Definite Integral
- Indefinite Integral
Definite Integral
In this integral, lower, and upper bound are defined i.e. (starts and end value). This integral is defined as:
∫ba f(x) dx = F(b) – F(a)
Where C is a constant number solution of the definite integral.
Indefinite Integral
In this integral, the lower and upper bounds are not defined. The integration of f(x) is F(x), and its formula can be represented as
∫f(x) dx = F(x) + C
Where C is an integrating constant that comes in indefinite integral.
Fundamental Theorems of Integral Calculus
In this theorem, f(x) is defined on the interval [a, b] and the given curve x = f(x) such that, a<x<b and x ε [a, b]. Then the integral ∫ba f(x) dx represents the area between intervals [a, b].
By this concept, we define the two theorems:
- First Fundamental Theorem of Integral
- Second Fundamental Theorem of Integral
1st Fundamental Theorem of Integral
It is defined as:
F(x) = ∫ba f(x) dx, for all x ≥ a
where the f(x) is defined on [a, b] and x ε [a, b]
2nd Fundamental Theorem of Integral
If f(x) is defined on the closed interval [a, b] and the other side F(x) that d/d(x) [ F(x)] = f(x) for all x.
The integral is defined as:
∫ba f(x) dx = F(b) – F(a) = L
Rules of Integration
There are some basic rules of integration discussed below in the table:
| Rules | General Formula |
| Sum Rule | ∫ [f(x) + g (x)] dx = ∫ [f(x)] dx+ ∫ [g(x)] dx |
| Difference Rule | ∫ [f(x) – g (x)] dx = ∫ [f(x)] dx – ∫ [g(x)] dx |
| Power Rule | ∫ [f(x)]n dx = [f(x)]n+1 / (n + 1) + c |
| Constant function Rule | ∫ [Q f(x)] dx = Q ∫ [f(x)] dx |
Formulas
Some useful trigonometric and exponential formulas are described in the following table.
| Function | Formula |
| Cos(x) | ∫ [cos(x)] dx = sin (x) + c |
| Sin (x) | ∫ [sin(x)] dx = -sin (x) + c |
| Tan(x) | ∫ [tan(x)] dx = -ln(cos(x)) + c |
| Sin-1(x) | ∫ dx/ (b2 – x2)1/2 = [ sin-1 (x/b)] + c |
| Cos-1(x) | ∫-dx/ (b2 – x2)1/2 = [ cos-1 (x/b)] + c |
| Tan-1(x) | ∫ dx/ (b2 + x2) = 1/b [ tan-1 (x/b)] + c |
| Exponential | ∫ enx dx = enx / n + c |
Examples Section
In this section, we will solve the integration with the help of examples.
Example 1
Find the integral of z2 (z – 1)
Solution:
Step 1: Let the given value is equal to A(z).
A(z) = z2 (z – 1)
A(z) = z3 – z
Step 2: Apply integral on both sides w.r.t the “z”
∫ A(z) dz = ∫ (z3 – z) dz
Step 3: Apply the Difference rule and separate the integral.
∫ [f(x) – g (x)] dx = ∫ [f(x)] dx – ∫ [g(x)] dx
∫ A(z) dz = ∫ z3 dz – ∫ z dz
Step 4: Solve the integral by power rule.
∫ [G(z)]n dz = [G(z)]n+1 / (n + 1) + C
∫ A(z) dz = z4 / 4 – z2/2 + C
Thus, ∫ z2 (z – 1) dz = (z4 / 4) – (z2/2) + C
An integration calculator is a helpful way for solving the problems of integral to avoid time-consuming calculations.
Example 2
Find the integral of 3tan(x) + 4cos(x).
Solution:
Step 1: Let the given value is equal to a function.
B(x) = 3tan(x) + 4cos(x)
Step 2: Apply the integral on both sides.
∫ B(x) dx = ∫ [3tan(x) + 4cos(x)]
Step 3: Separate the integral with the help of the sum rule.
∫ B(x) dx = ∫ [ 3 Tan(x)] dx + ∫ [4 cos(x)] dx
Step 4: Take out the Constant from the integral by constant function rule.
∫ B(x) dx =3 ∫ [tan(x)] dx + 4 ∫ [cos(x)] dx
Step 5: Apply the integral formula of “tan(x)” & “cos(x)”.
∫ B(x) dx = 3[ -ln(cos(x))] + 4 [ sin (x)] + c
∫ B(x) dx = – 3 ln(cos(x)) + 4 [ sin (x)] + c
∫ 3 Tan(x) + 4 cos(x) dx = -3 ln(cos(x)) + 4 [sin (x)] + c
Example 3
Find the solution of ∫20 (ex + x) dx
Solution:
Step 1: Let the given integral is equal to F(x).
E(x) = ∫20 (ex + x) dx
Step 2: Separate the integral by the sum rule.
∫ [h(x) + I(x)] dx = ∫ [h(x)] dx + ∫ [I(x)] dx
E(x) = ∫f0 (ex) dx + ∫20 (x) dx
Step 3: Apply the integration formula.
E(x) = ex [ex + (x2/2) |02
Step 4: Apply the definite integral definition { ∫ba f(x) dx = F(b)-F(a)} and simplify.
E(x) = [e2 – e0] + (1/2) [x2 – x0]
E(x) = [e2 – 1) + (1/2) [x2 – 1]
∫20 (ex + x) dx = [e2 – 1) + (1/2) [x2 – 1]
Summary
In this article, we discussed the basic detail of integral calculus with the help of the second fundamental theorem. Further, we have discussed some real-world examples, formulas, and rules. In the example section, we have discussed the methods to calculate the definite and indefinite integral by using the rules of integration.
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