Integrals
Imagine you’re filling up a swimming pool and want to know how much water it can hold. To find the total volume, you need to add up small parts, and this is where integrals come into play. In real life, integrals help us calculate areas, volumes, and other quantities by adding up infinitely small pieces. In integrals class 12, you’ll learn how to work with definite integrals, use properties of definite integrals, and apply integrals formulas to solve practical problems. In this article, we will explore integrals class 12, applications of integrals class 12, properties of definite integrals, integrals formulas, and class 12 integrals solutions to help you build a strong foundation in mathematics.
What are Integrals
Integrals are a way to add up small pieces to find the whole. For example, if you want to know how much water fills a pool or how much area is under a curve, integrals help you do that. It’s like taking tiny slices of a shape and adding them together to get the full picture. In math, when you know how something changes (its derivative), you can use integrals to find the original quantity or function.
Integrals are everywhere – from calculating distance traveled, to figuring out areas, and even in science and engineering problems. Once you understand how they work, you’ll see how useful they are in real-life situations!
Integral calculus is mainly divided into indefinite integrals and definite integrals.
Types of Integrals: Indefinite Integrals
Indefinite integrals are a way to find the original function when you know how it’s changing. It’s like working backwards from the derivative to get the function itself. This process is called indefinite integration, and it forms the basic definition of integration or antiderivatives.
Definition of Indefinite Integrals
If $\frac{d}{dx} F(x) = f(x)$, then we write: $\int f(x)dx = F(x) + C$
Here, $C$ is the constant of integration, and it shows that there are infinitely many solutions because you can choose any value for $C$. All these functions differ only by that constant.
Properties of Indefinite Integrals
(i) Inverse Relationship
Differentiation and integration are opposite processes:
$\frac{d}{dx}\int f(x)dx = f(x)$
$\int f'(x)dx = f(x) + C$
(ii) Family of Curves
Different integrals with the same derivative represent a group of curves that differ only by a constant. So, if two functions have the same derivative, their integrals are considered equivalent.
(iii) Sum Rule
You can break an integral of a sum into the sum of integrals:
$\int (f(x) + g(x))dx = \int f(x)dx + \int g(x)dx$
(iv) Constant Factor Rule
A constant can be taken out of the integral:
$\int a f(x)dx = a \int f(x)dx$
where $a$ is a constant.
Types of Integrals: Definite Integrals
While indefinite integrals give us a general solution, definite integrals help calculate the exact value of a function between two points. This is very useful in real-world problems where you need precise answers.
Definition of Definite Integrals
Let’s say $f(x)$ is a function defined on the interval $[a, b]$ and $F(x)$ is its antiderivative such that $\frac{d}{dx}F(x) = f(x)$. The definite integral is written as: $\int_a^b f(x)dx = F(b) - F(a)$
Here, $a$ is the lower limit and $b$ is the upper limit. This formula helps in calculating things like area under a curve or the total accumulated change in a quantity.
Applications of Inequalities in Definite Integration
Inequalities are useful to compare two functions, especially when solving problems related to application of integrals class 12.
1. Bounded Functions
If $m \leq f(x) \leq M$ for all $x$ in $[a, b]$, then: $m(b-a) \leq \int_a^b f(x)dx \leq M(b-a)$
2. Mean Value Theorem
If $f(x)$ is continuous on $[a, b]$, then there’s at least one $c \in [a, b]$ where: $f(c) = \frac{1}{b-a} \int_a^b f(x)dx$
3. Comparing Functions
If $f(x) \geq g(x)$ for all $x \in [a, b]$, then: $\int_a^b f(x)dx \geq \int_a^b g(x)dx$
4. Jensen’s Inequality
For convex functions $\phi(x)$ and a probability distribution: $\phi\left(\int_a^b f(x)dx\right) \leq \int_a^b \phi(f(x))dx$
5. Minkowski’s Inequality
For functions in $L^p$ space: $\left(\int_a^b |f(x)+g(x)|^p dx\right)^{1/p} \leq \left(\int_a^b |f(x)|^p dx\right)^{1/p} + \left(\int_a^b |g(x)|^p dx\right)^{1/p}$
Properties of Definite Integrals
The properties of definite integrals make it easier to solve problems in integrals class 12 and understand the behavior of functions across intervals.
$\int_a^b f(x)dx = \int_a^b f(t)dt$
$\int_a^b f(x)dx = -\int_b^a f(x)dx$ and $\int_a^a f(x)dx = 0$
$\int_a^b f(x)dx = \int_a^c f(x)dx + \int_c^b f(x)dx$
$\int_a^b f(x)dx = \int_a^b f(a+b-x)dx$
$\int_0^a f(x)dx = \int_0^a f(a-x)dx$
$\int_0^{2a} f(x)dx = \int_0^a f(x)dx + \int_0^a f(2a-x)dx$
- $\int_0^{2a} f(x)dx = \begin{cases}
2 \int_0^a f(x)dx, & \text{if } f(2a-x) = f(x) \\
0, & \text{if } f(2a-x) = -f(x)
\end{cases}$ - (i) $\int_{-a}^a f(x)dx = 2 \int_0^a f(x)dx$, if $f(x)$ is even, i.e., $f(-x) = f(x)$
(ii) $\int_{-a}^a f(x)dx = 0$, if $f(x)$ is odd, i.e., $f(-x) = -f(x)$
These properties are not just theoretical, they’re applied in many problems when working through class 12 integrals solutions or solving real-life problems using application of integrals class 12.
Difference between Definite and Indefinite Integrals
| Aspect | Definite Integrals | Indefinite Integrals |
|---|---|---|
| Definition | Finds the exact area or value between two points | Finds the general form of the original function |
| Limits | Has upper and lower limits (e.g., from $a$ to $b$) | No limits or boundaries |
| Result | Gives a numerical value | Gives a function with a constant $C$ |
| Application | Used to calculate area, volume, work, etc. | Used to find all possible antiderivatives |
| Example | $\int_a^b f(x)dx$ | $\int f(x)dx$ |
| Dependence on boundaries | Depends on the specific limits provided | Does not depend on limits |
| Interpretation | Represents accumulation or total change over an interval | Represents a family of functions differing by a constant |
Integrals Formulae
This section provides all the important formulas of integrals, including basic functions and rules, to help you solve problems easily and strengthen your understanding of integrals. These formulas are essential for both theory and practice in exams like JEE Main.
Indefinite Integral Formulae
We have provided below a list of important Indefinite Integral Formulas:
| Category | Function | Integral Formula |
|---|---|---|
| Algebraic Functions | $\int 0 ,dx$ | $c$, where $c$ is a constant |
| $\int k ,dx$ | $kx + c$, where $k$ is a constant | |
| $\int x^n ,dx$ | $\frac{x^{n+1}}{n+1} + c,\quad n \neq -1$ | |
| $\int \frac{1}{x} ,dx$ | $\log x$ | |
| Trigonometric Integrals | $\int \sin x ,dx$ | $-\cos x + c$ |
| $\int \cos x ,dx$ | $\sin x + c$ | |
| $\int \sec^2 x ,dx$ | $\tan x + c$ | |
| $\int \csc^2 x ,dx$ | $-\cot x + c$ | |
| $\int \sec x \tan x ,dx$ | $\sec x + c$ | |
| $\int \csc x \cot x ,dx$ | $-\csc x + c$ | |
| Exponential Functions | $\int e^x ,dx$ | $e^x + c$ |
| $\int a^x ,dx$ | $\frac{a^x}{\log a} + c$ | |
| Logarithmic Functions | $\int \frac{1}{x} ,dx$ | $\log x$ |
| Inverse Trigonometric Functions | $\int \frac{1}{\sqrt{1-x^2}} ,dx$ | $\sin^{-1}x + c$ |
| $\int \frac{1}{1+x^2} ,dx$ | $\tan^{-1}x + c$ |
Definite Integral Formulae
We have provided the formulae for definite integrals:
| Function / Rule | Definite Integral Formula |
|---|---|
| Constant function | $\int_a^b k ,dx = k(b - a)$ |
| Power function | $\int_a^b x^n ,dx = \frac{b^{n+1} - a^{n+1}}{n+1}, \quad n \neq -1$ |
| Reciprocal function | $\int_a^b \frac{1}{x} ,dx = \log \left |
| Sine function | $\int_a^b \sin x ,dx = -\cos b + \cos a$ |
| Cosine function | $\int_a^b \cos x ,dx = \sin b - \sin a$ |
| Secant squared function | $\int_a^b \sec^2 x ,dx = \tan b - \tan a$ |
| Cosecant squared function | $\int_a^b \csc^2 x ,dx = -\cot b + \cot a$ |
| Secant times tangent | $\int_a^b \sec x \tan x ,dx = \sec b - \sec a$ |
| Cosecant times cotangent | $\int_a^b \csc x \cot x ,dx = -\csc b + \csc a$ |
| Exponential function | $\int_a^b e^x ,dx = e^b - e^a$ |
| Exponential base $a$ | $\int_a^b a^x ,dx = \frac{a^b - a^a}{\log a}$ |
| Inverse square root | $\int_a^b \frac{1}{\sqrt{1-x^2}} ,dx = \sin^{-1} b - \sin^{-1} a$ |
| Inverse quadratic | $\int_a^b \frac{1}{1+x^2} ,dx = \tan^{-1} b - \tan^{-1} a$ |
| Property | Formula |
|---|---|
| Change of variable name | $\int_a^b f(x) ,dx = \int_a^b f(t) ,dt$ |
| Reversing limits | $\int_a^b f(x) ,dx = -\int_b^a f(x) ,dx$ |
| Integral over same limits | $\int_a^a f(x) ,dx = 0$ |
| Breaking into parts | $\int_a^b f(x) ,dx = \int_a^c f(x) ,dx + \int_c^b f(x) ,dx$ |
| Symmetry rule | $\int_a^b f(x) ,dx = \int_a^b f(a+b-x) ,dx$ |
| Even function rule | $\int_{-a}^a f(x) ,dx = 2\int_0^a f(x) ,dx,\quad$ if $f(-x) = f(x)$ |
| Odd function rule | $\int_{-a}^a f(x) ,dx = 0,\quad$ if $f(-x) = -f(x)$ |
Rules of Integration
In addition to memorizing formulas, understanding the rules of integration helps simplify problems and work with complex functions.
(a) Constant Multiple Rule
If $f(x)$ has an antiderivative, then multiplying it by a constant $k$ gives:
$\int kf(x)dx = k\int f(x)dx$
(b) Sum Rule
You can split the integral of a sum into the sum of integrals:
$\int (f(x) + g(x))dx = \int f(x)dx + \int g(x)dx$
(c) Difference Rule
Similarly, the difference between functions can be split:
$\int (f(x) - g(x))dx = \int f(x)dx - \int g(x)dx$
Integration by Parts
Integration By Parts Formula: Sometimes you’ll need to integrate products of functions. For this, we use integration by parts, a powerful technique in application of integrals class 12.
If two functions, $u(x)$ and $v(x)$, are given, then:
$\int u v ,dx = u \times \int v ,dx - \int \left(\frac{du}{dx} \times \int v ,dx \right)dx$
This method helps when simple formulas aren’t enough, especially in problems involving products of algebraic, logarithmic, or trigonometric functions.
ILATE Rule – Choosing $u$
When selecting which function to assign as $u$, we use the ILATE rule:
Inverse > Logarithmic > Algebraic > Trigonometric > Exponential
For example, if your integral includes both a logarithmic and an algebraic function, choose the logarithmic function as $u$ because it comes first in ILATE.
Applications of Integrals Class 12
Integrals are more than just formulas — they are essential tools that help solve real problems in mathematics. Below are some of the key areas where integrals are used, along with simple explanations to help you understand their importance.
1. Finding Area Under Curves
This is one of the most common uses of integrals. With definite integrals, you can calculate the exact area between a curve and the x-axis over a given interval.
For example, to find the area under the curve defined by the function $y = x^2$ from $x = 1$ to $x = 3$, you would calculate: $\int_1^3 x^2 dx$
This method is widely applied in geometry and algebra problems.
2. Calculating Volumes of Solids of Revolution
Integrals help you find the volume of 3D objects created by rotating a curve around an axis.
For example, the volume of a solid formed by rotating $y = f(x)$ around the x-axis from $x = a$ to $x = b$ is: $V = \pi \int_a^b [f(x)]^2 dx$
This is useful in geometry and engineering applications.
3. Finding Length of Curves
Integrals are used to calculate the length of a curve between two points, which helps in areas like design, architecture, and engineering.
The formula for the length of a curve is: $L = \int_a^b \sqrt{1 + \left(\frac{dy}{dx}\right)^2} dx$
4. Calculating Work Done by a Force
In mathematics, integrals are used to find the total work done when a force varies across a distance.
For example: $W = \int_a^b F(x) dx$
This application connects mathematics with physics and engineering.
5. Finding the Average Value of a Function
Integrals can be used to find the average value of a function over an interval, which is useful in statistics and analysis.
The formula is: $\text{Average} = \frac{1}{b-a} \int_a^b f(x) dx$
6. Solving Differential Equations
Integrals help in finding solutions to equations that describe how quantities change. These are known as differential equations.
For example, if $\frac{dy}{dx} = f(x)$, then the solution is: $y = \int f(x) dx + c$
This is widely applied in mathematical modeling and real-world problems.
7. Finding Moments and Centroids
Integrals are used to calculate the center of mass or centroid of an object by integrating over its shape.
For example: $M_x = \int y f(x) dx$
$M_y = \int x f(x) dx$
These calculations are important in mechanics and geometry.
8. Probability and Statistics
Integrals are used to calculate probabilities when dealing with continuous random variables by integrating probability density functions.
For example: $P(a \leq x \leq b) = \int_a^b f(x) dx$
This forms the mathematical foundation for probability and statistical analysis.
9. Solving Problems Involving Rate of Change
Integrals are used wherever you need to calculate accumulated quantities like distance, growth, or decay over time.
By reversing the process of differentiation, integrals allow you to find the total amount of something from its rate of change.
Why These Applications Matter
Integrals help you solve real-world problems in geometry, physics, engineering, economics, and more.
They provide exact answers rather than approximations.
They form the basis for advanced mathematical techniques like differential equations and statistical analysis.
Mastering integrals allows you to understand how mathematical principles apply to everyday situations.
List of Topics related to Integrals according to NCERT/JEE MAIN
This section covers all the important topics on integrals that are part of the NCERT syllabus and frequently appear in the JEE Main exam.
Important Books and Resources for Class 12 Integrals
Here, you’ll find recommended books and study materials that help build a strong understanding of integrals and prepare effectively for exams.
| Book Title | Author / Publisher | Description |
|---|---|---|
| NCERT Class 12 Mathematics | NCERT | Official textbook with fundamental theory and exercises on integrals. |
| Mathematics for Class 12 | R.D. Sharma | Detailed explanations, solved examples, and practice on integration techniques. |
| Objective Mathematics | R.S. Aggarwal | Provides practice problems and objective questions on integrals. |
| Arihant All-In-One Mathematics | Arihant | Comprehensive theory and unsolved/specially solved integration problems for JEE and board exams. |
| Integral Calculus | M.L. Khanna | Advanced textbook covering integral calculus concepts and applications. |
NCERT Resources
This section lists NCERT textbooks and reference materials that explain integrals clearly and form the foundation for solving related problems.
NCERT Subjectwise Resources
Explore subject-wise NCERT resources that cover different study materials such as notes, solutions and exemplar problems to support your preparation in a structured way.
Practice Questions based on Integrals
This section includes important practice questions and problems based on integrals to help you test your understanding and improve problem-solving skills.
Frequently Asked Questions (FAQs)
An integral is a mathematical concept that helps you find the area under a curve, total quantity from rates of change, and solve problems involving accumulation. It’s important because it’s widely used in physics, engineering, economics, and other fields.
A definite integral gives you a numerical value representing area or total change between two points, while an indefinite integral gives you a general formula or family of functions, including a constant of integration.
Integrals are the reverse process of derivatives. If differentiation finds how a quantity changes, integration helps to find the original quantity from its rate of change.
The constant of integration ($C$) is added because integration gives a family of solutions. Different constants correspond to different curves that have the same derivative.
When a function f(x) is integrated, you obtain antiderivative F(x). You may also acquire the area under the curve by evaluating the anti-derivative over a given domain [a, b].