Integrals

Have you ever wondered how a roller coaster is in a shape that withstands the pressure from the speed and weight??? It is all because of calculus. In this case, calculus, especially integration, is used to design the shape of the roller coaster such that it can handle the stress from the forces exerted.

This article is about the concept of class 12 Integrals. Integrals chapter is not only essential for board exams but also for competitive exams like the Joint Entrance Examination (JEE Main), and other entrance exams such as SRMJEE, BITSAT, WBJEE, VITEEE, BCECE, and more.

Integrals

One cannot imagine a world without differentiation and integration. In this century, we witnessed remarkable scientific advancement owing to the application of these two basic components of Mathematics. Calculus serve as unavoidable tool for finding solutions to the variety of problems that arise in physics, astronomy, engineering, chemistry, geology, biology, and social sciences. The main application of integrals is finding the area bounded under a curve.

Introduction to Integrals Class 12

Integration is the reverse process of differentiation. In integration, we find the function whose differential coefficient is given. The rate of change of a quantity y concerning another quantity x is called the derivative or differential coefficient of y concerning x. Geometrically, the Differentiation of a function at a point represents the slope of the tangent to the graph of the function at that point.

Integrals are based on a limiting procedure which approximates the area of a curvilinear region by breaking the region into thin vertical slabs.

For example,

$\begin{aligned} & \frac{d}{d x}(\sin x)=\cos x \\ & \frac{d}{d x}\left(x^2\right)=2 x \\ & \frac{d}{d x}\left(e^x\right)=e^x\end{aligned}$

In the above example, the function $\cos (x)$ is the derivative of $\sin (x)$. We say that $\sin (x)$ is an antiderivative (or an integral) of $\cos (x)$. Similarly, $x^2$ and $e^x$ are the antiderivatives (or integrals) of $2 x$ and $\mathrm{e}^{\mathrm{x}}$ respectively.

Also note that the derivative of a constant (C) is zero. So we can write the above examples as:

$\begin{aligned}
& \frac{d}{d x}(\sin x+c)=\cos x \\
& \frac{d}{d x}\left(x^2+c\right)=2 x \\
& \frac{d}{d x}\left(e^x+c\right)=e^x
\end{aligned}$

Thus, the anti-derivatives (or integrals) of the above functions are not unique. There exist infinitely many anti-derivatives of each of these functions which can be obtained by selecting C arbitrarily from the set of real numbers.

For this reason, C is referred to as an arbitrary constant. C is the parameter by varying which one gets different anti-derivatives (or integrals) of the given function.

Integral calculus is mainly divided into indefinite integrals and definite integrals.

Indefinite Integrals

Indefinite integrals are process of obtaining the function from the derivative. Indefintie integration is the basic definition of integration(antiderivative). Let $\frac{d}{d x} \mathrm{~F}(x)=f(x)$. Then, we write $\int f(x) d x=\mathrm{F}(x)+\mathrm{C}$. These integrals are called indefinite integrals or general integrals, $C$ is called a constant of integration. All these integrals differ by a constant.

Properties of Indefinte Integrals

(i) The process of differentiation and integration are inverse of each other, i.e., $\frac{d}{d x} \int f(x) d x=f(x)$ and $\int f^{\prime}(x) d x=f(x)+\mathrm{C}$, where C is any arbitrary constant.

(ii) Two indefinite integrals with the same derivative lead to the same family of curves and so they are equivalent. So if $f$ and $g$ are two functions such that $\frac{d}{d x} \int f(x) d x=\frac{d}{d x} \int g(x) d x$, then $\int f(x) d x$ and $\int g(x) d x$ are equivalent.

(iii) The integral of the sum of two functions equals the sum of the integrals of the functions i.e., $\int(f(x)+g(x)) d x=\int f(x) d x+\int g(x) d x$.

(iv) A constant factor may be written either before or after the integral sign, i.e., $\int a f(x) d x=a \int f(x) d x$, where ' $a$ ' is a constant.

Definite Integrals

Definite integrals is used to calculate the value of function over a specific interval.

Let f be a function of x defined on the closed interval [a, b] and F be another function such that $\frac{d}{d x}(F(x))=f(x)$ for all $x$ in the domain of $f$, then

$\int_a^b f(x) d x=[F(x)+c]_a^b=F(b)-F(a)$ is called the definite integral of the function $f(x)$ over the interval $[a, b]$, where $a$ is called the lower limit of the integral and $b$ is called the upper limit of the integral.

Applications of inequalities in Definite Integration

Inequalities gives relationship between two expressions that are not equal to one another. Symbols denoting the inequalities are <, >, ≤, ≥, and ≠.

Inequalities are used in definite integration to show compare to functions.

Results with inequalities in definite integration

1. If $m \leq f(x) \leq M$ for all $x$ in the interval $[a, b]$, then:

$
m(b-a) \leq \int_a^b f(x) d x \leq M(b-a)
$

2. Mean Value Theorem

If $f(x)$ is continuous on $[a, b]$, there exists a $c \in[a, b]$ such that:

$
f(c)=\frac{1}{b-a} \int_a^b f(x) d x
$

3. If $f(x) \geq g(x)$ for all $x \in[a, b]$, then:

$
\int_a^b f(x) d x \geq \int_a^b g(x) d x
$

This result is used to compare two functions.

4. For convex functions $\phi(x)$ and a probability distribution $p(x)$, Jensen's inequality states that:

$
\phi\left(\int_a^b f(x) d x\right) \leq \int_a^b \phi(f(x)) d x
$

5. Minkowski's Inequality

For $f(x)$ and $g(x)$ both in $L^p$ space:

$
\left(\int_a^b|f(x)+g(x)|^p d x\right)^{1 / p} \leq\left(\int_a^b|f(x)|^p d x\right)^{1 / p}+\left(\int_a^b|g(x)|^p d x\right)^{1 / p}
$

Properties of Definite Integrals

The properties of definte integrals are,

1. $\int_a^b f(x) d x=\int_a^b f(t) d t$

2. $\int_a^b f(x) d x=-\int_b^a f(x) d x, \text { in particular, } \int_a^a f(x) d x=0$

3. $\int_a^b f(x) d x=\int_a^c f(x) d x+\int_c^b f(x) d x$

4. $\int_a^b f(x) d x=\int_a^b f(a+b-x) d x$

5. $\int_0^a f(x) d x=\int_0^a f(a-x) d x$

6. $\int_0^{2 a} f(x) d x=\int_0^a f(x) d x+\int_0^a f(2 a-x) d x$

7. $\int_0^{2 a} f(x) d x=\left\{\begin{array}{l}
2 \int_0^a f(x) d x, \text { if } f(2 a-x)=f(x), \\
0, \text { if } f(2 a-x)=-f(x) .
\end{array}\right.
$

8. (i) $\int_{-a}^a f(x) d x=2 \int_0^a f(x) d x$, if $f$ is an even function i.e., $f(-x)=f(x)$
(ii) $\int_{-a}^a f(x) d x=0$, if $f$ is an odd function i.e., $f(-x)=-f(x)$

Integrals Formula

Class 12 Integrals formula include integration formulas of basic polynomial functions, trigonometric functions, exponential functions etc. and rules of differentiation.

Indefinite Integrals Formulas

1. $\int 0 d x=c$, where $c$ is a constant

2. $\int k d x=k x+c$ where $c$ is a constant

3. $\int x^n d x=\frac{x^{n+1}}{n+1}+c, \quad n \neq-1$ (Power rule)

4. $\int \frac{1}{x} d x=\log |x|+c$

5. $\int \sin x d x=-\cos x+c$

6. $\int \cos x d x=\sin x+c$

7. $\int \sec ^2 x d x=\tan x+c$

8. $\int \operatorname{cosec}^2 x d x=-\cot x+c$

9. $\int \sec x \tan x d x=\sec x+c \\$

10. $\int \operatorname{cosec} x \cot x d x=-\operatorname{cosec} x+c $

11. $\int e^x d x=e^x+c$

12. $\int a^x d x=\frac{a^x}{\log a}+c$

13. $\int \frac{1}{\sqrt{1-x^2}} d x=\sin ^{-1} x+c$

14. $\int \frac{1}{1+x^2} d x=\tan ^{-1} x+c$

Rules of Integration

$f(x)$ and $g(x)$ are functions with antiderivatives $∫ f(x) dx$ and $∫ g(x) dx$. Then,

(a) $\int \mathrm{kf}(x) d x=k \int f(x) d x$ for any constant $k$.

(b) $\int(f(x)+g(x)) d x=\int f(x) d x+\int g(x) d x$

(c) $\int(f(x)-g(x)) d x=\int f(x) d x-\int g(x) d x$

(d) Integration of parts: To evaluate the integration of a function which is a product of two functions, we use the method of integration by parts.

If two functions of $x, u, and v$ are given,

where $u$ is function of $u(x)$,

$v$ is function of $v(x)$

then

$\int u v d x=u \times \int v d x-\int\left\{\frac{d u}{d x} \int v d x\right\} d x$

Usually, we use the following preference order for selecting the first function. (Inverse, Logarithmic, Algebraic, Trigonometric, Exponential). In the above-stated order, the function on the left is always chosen as the first function. This rule is called ILATE.

For example, if an integral contains a logarithmic function and an algebraic function, we should choose the logarithmic function as the first function ($u$).

Applications of Integrals Class 12

Now, let us look into some of the application of integrals class 12.

• One of the important application of integrals is finding the area bounded by two or more curves. Integrals are also used to calculate the quantities of velocity, acceleration and distance. These are mainly used in physics and engineering.
• The application of integral by finding area bounded by curves is also used in economics to find the total revenue and cost over time.
• Integrals are used to form images which has greater applications in technology. This is one of the important applications which helps in the field of medicine through CT and MRI scans.
• In the topic of probability, the integrals are in the concept of probability density function(PDF).

List of Topics According to NCERT/JEE MAIN

Importance of Class 12 Integrals

Integrals have a significant weighting in the IIT JEE test, which is a national level exam for 12th grade students that aids in admission to the country's top engineering universities. It is one of the most difficult exams in the country, and it has a significant impact on students' futures. Several students begin studying as early as Class 11 in order to pass this test. When it comes to math, the significance of these chapters cannot be overstated due to their great weightage. You may begin and continue your studies with the standard books and these revision notes, which will ensure that you do not miss any crucial ideas and can be used to revise before any test or actual examination.

How to Study Class 12 Integrals?

Start preparing by understanding and practicing the concept of limits. Try to be clear on the definition, rules and formulas of indefinite integrals and definite integrals. Practice many problems from each topic for better understanding. Practice integrals class 12 previous year questions.

If you are preparing for competitive exams then solve as many problems as you can. Do not jump on the solution right away. Remember if your basics are clear you should be able to solve any question on this topic.

NCERT Notes Subject wise link:

• NCERT Notes for class 12 Physics.
• NCERT Notes for class 12 Chemistry.
• NCERT Notes for class 12 Mathematics.
• NCERT Notes for class 12 Biology

Important Books for Class 12 Integrals

Start from NCERT Books, the illustration is simple and lucid. You should be able to understand most of the things. Solve all problems (including miscellaneous problem) of NCERT. If you do this, your basic level of preparation will be completed.

Then you can refer to the book Amit M Aggarwal's differential and integral calculus or Cengage Algebra Textbook by G. Tewani but make sure you follow any one of these not all. Integrals are explained very well in these books and there are an ample amount of questions with crystal clear concepts. Choice of reference book depends on person to person, find the book that best suits you the best, depending on how well you are clear with the concepts and the difficulty of the questions you require.

NCERT Solutions Subject wise link:

• NCERT solutions for class 12 Physics.
• NCERT solutions for class 12 Chemistry.
• NCERT solutions for class 12 Mathematics.
• NCERT solutions for class 12 Biology.

NCERT Exemplar Solutions Subject wise link:

• NCERT exemplar solutions for class 12 Physics.
• NCERT exemplar solutions for class 12 Chemistry.
• NCERT exemplar solutions for class 12 Mathematics.
• NCERT exemplar solutions for class 12 Biology.