Integral Formulas
In mathematics, an integral is the mathematical process of calculating the area under a curve. This guide will walk you through the different types of integrals, how to calculate them, and when to use them.
What is an Integral?
An integral is a mathematical function that can be used to calculate the area under a curve. It can also be used to calculate the volume of a three-dimensional object. In calculus, an integral is a way to find the area under a curve or the volume of a three-dimensional object without having to break it up into pieces.
The Three Different Types of Integrals
Integrals are a fundamental part of calculus, and they have many applications in mathematics and physics. There are three main types of integrals: definite, indefinite, and improper. Each type has its own set of rules and properties, and knowing when to use each one is essential for solving problems correctly.
Definite integrals are used to find the value of a function over a specific interval. Indefinite integrals are used to find the general form of a function, without reference to any particular interval. Improper integrals are used to find the value of a function that doesn’t have a definite value (for example, a function that approaches infinity as x approaches a certain point).
Knowing which type of integral to use is half the battle – the other half is actually computing the integral itself. Luckily, there are some helpful formulas that can make this task much easier. In this guide, Knowledge Glow will go over some of the most commonly used integral formulas, so you can learn how to solve problems quickly and efficiently.
The Process of Integration
Integration is the process of finding the area under a curve. This can be a very daunting task, but with the help of integral formulas, it can be much easier. There are a few steps that need to be taken in order to find the area under a curve. The first step is to find the function that will be integrated. This is usually done by taking the derivative of the original function. The next step is to find the limits of integration. These are usually the points where the function changes from positive to negative, or vice versa. After the limits have been found, the next step is to choose a method of integration. There are many different methods, but the most common are u-substitution and integration by parts. Once a method has been chosen, it is simply a matter of following the steps and integrating the function.
How to Use Integrals in Real-World Situations
Integrals are a powerful tool that can be used to solve many real-world problems.
1. Finding the area of a region:
One common use of integrals is to find the area of a region. To do this, we must first identify the boundaries of the region. For example, let’s say we want to find the area of the region bounded by the x-axis, the y-axis, and the line y=x^2. We can see that the boundaries of this region are:
The x-axis: x=0
The y-axis: y=0
The line y=x^2: y=x^2
Now that we have identified the boundaries of the region, we can set up our integral. We will use the formula for finding the area of a region bounded by two curves:
Area = ∫bdy – ∫ady
Where b and a are the upper and lower bounds of the integral, respectively, and y is the equation of the boundary curve. In our case, we have:
Basic Integral Formulas List
List of basic integral formulas are :
- ∫ 1 dx = x + C
- ∫ a dx = ax+ C
- ∫ xn dx = ((xn+1)/(n+1))+C ; n≠1
- ∫ sin x dx = – cos x + C
- ∫ cos x dx = sin x + C
- ∫ sec2x dx = tan x + C
- ∫ csc2x dx = -cot x + C
- ∫ sec x (tan x) dx = sec x + C
- ∫ csc x ( cot x) dx = – csc x + C
- ∫ (1/x) dx = ln |x| + C
- ∫ ex dx = ex+ C
- ∫ ax dx = (ax/ln a) + C ; a>0, a≠1
Also Read: Important Maths Formulas for Class 8
Some other important integration formulas
Main Classification of Integral Formulas
Above listed integral formulas are classified based on below following functions.
- Rational functions
- Trigonometric functions
- Irrational functions
- Inverse trigonometric functions
- Inverse hyperbolic functions
- Hyperbolic functions
- Exponential functions
- Gaussian functions
- Logarithmic functions
Click here to download the PDF of important integral formulas.
Conclusion
This concludes our guide on integral formulas. We hope that you found this information to be helpful and that you now feel more confident in using integrals to solve problems. Remember, practice makes perfect, so keep at it and you’ll be a master of integrals in no time!
