Integral Calculus – Definition, Method, and Examples

Integral calculus is a crucial branch of calculus in which we study the concept of integrals, their properties as well as their applications. Finding the integral of the function is referred to as integration. Integration serves as the inverse operation of differentiation. The integration allows us to calculate the net change of a quantity over a specified interval. Integrals can be used to calculate areas under curves, calculate solid volumes, and solve differential equations.

In this article, we will explore the definition, properties, application, and solved examples of integral calculus.

Definition and Explanation of Integral Calculus

Integral calculus involves using integration to calculate the area under a curve. Integration finds the antiderivative of a function and is represented as ∫. If f is a differentiable function and f' is its derivative, integration of f' gives us f. This process helps us uncover the original function from its derivative.

Integral has two main types:

1. Definite integral

2. Indefinite integral

Definite Integral:

The definite integral is used to calculate the area under the curve of the function within a specific interval. The definite integral of a function f(x) over the interval [a, b] is denoted as

∫ab f(x) dx

This calculation gives the net area between the curve of the function and the x-axis within the interval [a, b].

Indefinite Integral

The indefinite integral is also known as an antiderivative. The indefinite integral represents a general function whose derivative is the given function. The indefinite integral of a function f(x) is denoted as

∫f(x) dx + C

Where the right hand side of the equation signifies the integral of f(x) with respect to x.

Here is a breakdown of the terms involved:

• F(x) is referred to as the antiderivative or primitive function.

• f(x) is called the integrand, which is the function being integrated.

• dx is referred to integrating agent. It indicates the independent variable with respect to which the integration is being performed.

• C is the constant of integration.

• x represents the variable of integration.

Fundamental Theorems of Calculus

The Fundamental Theorems of Calculus are two crucial principles in calculus that link the concepts of differentiation and integration.

First Fundamental Theorem of Calculus:

It states that if f(x) is a continuous function on a closed interval [a, b] and F(x) is an antiderivative of f(x), then the definite integral of f(x) from a to b is equal to the difference between the values of F(x) at b and a:

∫a b f(x) dx = F (b) – F (a)

Second Fundamental Theorem of Calculus:

It states that if f(x) is a continuous function on an interval [a, b] and F(x) is any antiderivative of f(x), then the derivative of the integral of f(x) from a to x is f(x).

Let F(x) = ∫a b f(x) dx

d/dx F(x) = d/dx (∫a b f(x) dx) ⇒ F’(x) = f(x)

Some Formulas for Integral

Here is a list of common formulas for finding integrals. You will become more adept at solving integration problems as you learn these formulas.

Properties of Definite Integral

Property Name Expression Explanation

Sum/ Difference ∫ab (f(x) ± g(x)) dx = ∫ab f(x) dx ± ∫ab g(x) dx The integral of a sum or difference is the sum or difference of the integrals.

Constant Multiple ∫ (k. f(x)) dx = k. ∫ f(x) dx A constant can be factored out of the integral.

Reverse Interval ∫ab f(x) dx = –∫ba f(x) dx

 Reversing the limits changes the sign of the integral.

Zero-length interval ∫aa f(x) dx = 0 When the interval has zero length, the integral is zero

Combining Intervals ∫ac f(x) dx + ∫cb f(x) dx = ∫ab f(x) dx The integral over a combined interval is the sum of the integrals over its subintervals.

Techniques for Finding Integral

There are several methods used for finding the integral of function in calculus. Some commonly used techniques are given below:

• Integration by Substitution

• Integration by part

• Partial fraction Decomposition

Integration by Substitution:

Substitution involves replacing a complicated expression with a simpler one by introducing a new variable. This technique is particularly effective for composite functions and expressions involving chains of functions. If s is the function of x, then

s’ = ds/dx

∫ f(s) s’ dx = ∫ f(s) du, where s = g(x)

Integration by Parts:

This technique is based on the product rule for differentiation. It is useful for integrating products of two functions. The formula is

∫ (f(x) g(x)) dx = f(x) ∫ g(x) dx – ∫ (f’(x) ∫ g(x) dx) dx

Partial Fraction Decomposition:

An easier way to integrate a complex rational function is to break it down into simpler fractions. It is especially helpful for integrating functions with denominators that can be factored.

An integral calculator with steps is an online tool that helps you solve integral calculus problems using the above-mentioned techniques and properties.

Solved Examples of Integral Calculus

Here are some solved examples of the integral calculus:

Example 1:

Evaluate the indefinite integral ∫(x3 + 2x - 5) dx.

Solution:

Integrate term by term

The integral of xn is xn+1/n+1

∫x3 dx = (1/4) x4

A constant can be factored out of the integral.

∫2x dx = 2 ∫x dx

The integral of xn is xn+1/n+1

∫2x dx = x2

The integral of constant a is ax

∫(-5) dx = -5x

Combine the results: ∫(x^3 + 2x - 5) dx = (1/4) x4 + x2 - 5x + C.

Thus, ∫(x3 + 2x - 5) dx = (1/4) x4 + x2 - 5x + C.

Example 2:

Evaluate ∫0𝜋 tan (x) dx

Solution

As we know that

∫ tan (x) = ln |sec(x)| + C

∫0𝜋 tan (x) dx = [ln |sec(x)|]0𝜋

= ln |sec (𝜋)| – ln |sec (0)|

= ln | (–1)| – ln |1|

= ln (1) – ln (1) = 0

Thus, ∫0𝜋 tan (x) dx = 0

Conclusion

In this article, we explain the definition of integral calculus and its types. We delve into the fundamental theorems of calculus that establish a link between integration and differentiation. We also discuss some common properties of the definite integral.

Additionally, we cover applications of integral calculus. To help our readers understand this concept better, we provide several solved examples of integrals.