# Integrals Class 12 Notes Maths Chapter 7 - CBSE

## Chapter:7

## What are Integrals ?

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## Integration

The process of integration is an opposite process to

that of differentiation.

∫ f (x) dx = F(x) + C

Where ∫ —› symbol of integration

f(x) —› integrand

dx —› element of integration

F —› primitive

C —› constant of integration In differential calculus, we are given a function and we are required to differentiate it, but in integral calculus we required to find a function whose differential coefficient (or derivative) is given.

Hence if,

d(F(x) + c) = f(x)/dx

We write that, ∫ f(x)dx = F(x) + c

## Standard Integrals

$$\text{(i)\space }\int 0\space \text{dx}=\text{C(constant)}\\ \text{(ii)}\int 1\text{dx} = x+\text{C}\\ \text{(iii)}\int x^n\text{dx}= \frac{x^{n+1}}{n+1}+\text{C} \space\text{if} ≠ -1\\ \text{(iv)} \int\frac{1}{x}\text{dx}=\text{log}_e\text{x}+\text{C}\\ \text{(v)}\int e^x\text{dx}=e^x+\text{C}$$

$$\text{(vi)}\int a^x\text{dx}=\frac{a^x}{\text{log}_ea}+\text{C}\\ \text{(vii)} \int \text{sin}\space x \space \text{dx\space=\space -cos}\space x+\text{C}\\ \text{(viii)}\int \text{cos}\space x \space\text{dx}=\text{sin}\space x+\text{C}\\ \text{(ix)}\int \text{sec}^2x\space\text{dx}= \text{tan}\space x+\text{C}\\ \text{(x)}\int\text{cosec}^2x \space \text{dx} =-\text{cot}\space x+\text{C}$$

$$\text{(xi)}\int \text{sec}\space x .\text{tan}\space x\space \text{dx}=\text{sec}\space x+\text{C}\\ \text{(xii)} \int \text{cosec}\space x\space.\space \text{cot}\space x\space\text{dx} =-\text{cosec x}+\text{C}\\\text{(xiii)}\int \text{cot}\space x\space\text{dx} = \text{log}|\text{sin} \space x|+\text{C}\\ \text{(xiv)}\int\text{tan}\space x \space\text{dx}=-\text{log}|\text{cos}\space x|+\text{C}\\\text{(xv)}\int\text{sec}\space x\space\text{dx}= \text{log}|\text{sec}\space x + \text{tan \space x}|+\text{C}=\text{log}|\text{tan}(\frac{\pi}{2}+\frac{x}{2})|+ \text{C}\\ \text{(xvi)} \int \text{cosec}\space x\space\text{dx}=\text{log}|\text{cosec}\space x-\text{cot}\space x|=\text{log}|\text{tan} \frac{x}{2}|+\text{C}$$

$$\text{(xvii)}\int\frac{1}{\sqrt{1-x^2}}\text{dx}=\text{sin}^{-1}x+\text{C}=-\text{cos}^{-1}x+\text{C}\\\text{(xviii)}-\int\frac{1}{\sqrt{1-x^2}}\text{dx}=\text{cos}^{-1}x+\text{C}=-\text{sin}^{-1}x+\text{C}\\\text{(xix)}\int\frac{\text{dx}}{x\sqrt{x^2-1}}=\text{sec}^{-1}x+\text{C}=-\text{cosec}^{-1}x+\text{C}\\\text{(xx)}-\int\frac{1}{x\sqrt{x^2-1}}\text{dx}=\text{cosec}^{-1}x+\text{C}=-\text{sec}^{-1}+\text{C}\\\text{(xxi)}\int\frac{1}{1+x^2}\text{dx}=\text{tan}^{-1}x+\text{C}=-\text{cot}^{-1}x+\text{C}\\\text{(xxii)}\int\frac{1}{1+x^2}\text{dx}=\text{cot}^{-1}x+\text{C}=-\text{tan}^{-1}x+\text{C}$$

## Methods Of Integration

We need to develop additional techniques or methods for finding the integrals by reducing them into standard forms. Here are some important methods:

A. Integration by Substitution

B. Integration by Partial Fraction

C. Integration by Parts

## A. Integration by Substitution

^{2}x = 1-cos 2x or 2cos

^{2}x = 1 + cos 2x if n is an even positive integer. (c) For evaluating integrals ∫ secn x dx or ∫ cosecn x dx where n = positive integer, (i) Put tan x = t, if the index of sec x is a positive even integer. (ii) Put cot x = t, if the index of cosec x is a positive even integer.

$$\text{(d)}\space \text{(i)}\space\text{If}\space \sqrt{a^2-x^2}\space\text{occurs in the integrand, put x = a sin}\space \alpha.\\ \qquad \text{(ii)}\space\text{If}\sqrt{a^2+x^2}\space\text{occurs in the integrand, put x = a tan} \space \alpha.\\ \qquad\text{(iii) If}\sqrt{x^2-a^2}\space\text{occurs in the integrand, put x = a sec}\space \alpha.$$

**B. Partial fractions**

Let f(x) =g(x)/h(x) be a proper rational function. First of all, we split up the denominator h(x) as the product of non-repeated linear or repeated linear factors or non-repeated or repeated quadratics (which cannot be split into real linear factors). Then f(x) can be written as the sum of the fractions in which the numerator is either a constant or real linear polynomials. For this the following points should be kept in mind.

(a) Corresponding to each non-repeated linear factor a x+ b, there is a partial fraction of the form A/(ax+b)

^{2}

^{2}+bx+c

^{2}+bx+c) +(Cx + D)/(ax

^{2}+ bx + c)

^{2}

^{2}= y the resulting algebraic function in y into partial fraction and then put y = x

^{2}in the resulting partial fraction.

**C. Integration By Parts**

Consider u and v as two functions of x, then integration

by parts says that; ∫ u.v dx = u { ∫ v dx} - ∫{du/dx. ∫ v dx } dx Where, u is known as first function and

v is known as second function.

v is known as second function.

i.e., Integral of product of two = (first function *

(integral of second function) - integral of

{(differentiation of first function) * (integral of

second function)}

We decide the function as first which comes first in

order from top to bottom.

INVERSE —› I

LOGARITHMIC —› L

ALGEBRIC —› A

TRIGONOMETRIC —› T

EXPONENTIAL —› E

Or which comes first in word 'ILATE'.

## Fundamental Theorem Of Calculus

The fundamental theorem of calculus is a theorem that links the concept of differentiating a function with the concept of integrating a function. The two operations are inverses of each. other apart from a constant value which depends where one starts to compute area. The first part of the theorem, called first fundamental

theorem of calculus. Let f be a continuous real-valued function defined on a closed interval [a,b] and the function F(x) is defined by

$$\text{F(x)} = \int^x_a \text{f(t)}\space \text{dt}$$

then F’(x) = f(x)

The second part of the theorem called second fundamental theorem of calculus. Let f be a continuous real-valued function on a closed interval [a,b], then

$$ \int^b_a \text{f(x)}\text{dx}= \text{F(b)}-\text{F(a)}$$

## Definite Integral

The definite integral has a unique value. A definite integral is denoted by

$$ \int^b_a \text{f(x)}\text{dx},$$

where a is called the lower limit of the integral and b is called the upper limit of the integral.

## Standard Integrals

$$\text{(i)} \int^b_a \text{f(x)}\text{dx}=\text{F(b)\space-\space F(a)}\space \text{where}\int \text{f(x)dx=\space \text{F(x)\space +\space\text{C}}}\\\text{(ii)} \int^b_a\text{f(x)dx}=\int^b_a\text{f(t)}\text{dt}\\\text{(iii)}\int^b_a \text{f(x)}\text{dx}=-\int^a_b \text{f(x)dx}\\\text{(iv)}\int^b_a\text{f(x)dx} = \int^c_a\text{f(x)dx}\space+\space \int^b_c\text{f(x)dx},\\ \text{where the point c lies between a and b}\\\text{(v)}\int^b_a\text{f(x)dx}=\int^b_a\text{f(a\space+\space b-x)dx}\\\text{(vi)}\int^a_0 \text{f(x)dx}=\int^a_0 \text{f(a-x)dx}\\\text{(vii)}\int^\text{2a}_0\text{f(x)dx}=\int^a_0\text{{f(x)dx + f(2a-x)}}\text{dx}\\\text{(viii)} \int^{2a}_0\text{f(x)dx}= 2\int^a_0\text{f(x)dx,}\space \text{if}\space \text{f(2a-x)}= \text{f(x)}\space \text{and} \space 0, \text{if} \space \text{f(2a-x)}=-\text{f(x)}\\ \text{(ix)}\int^a_{-a} \text{f(x)dx}=2\int^a_0 \text{f(x)dx,}\space \text{if} \space \text{f(-x)}=\text{f(x)} \space \text{and} \space 0, \text{if f(-x) = -f(x)}$$

## Evaluation Of Definite Integrals By Substitution

$$\text{To evaluate}\int^b_a \text{f(x)dx,} \space \text{by substitution, the following steps are useful}$$

- Consider the integral without limits and substitute, y=f(x) or x=g(y) to reduce the given integral to a known form.
- lntegrate the new integrand with respect to the new variable without mentioning the constant of integration.
- Resubstitute for the new variables and write the answer in terms of the original variable.
- Find the values of answer obtained in III step at he given limits of integral and find the difference of the values at the upper and lower limits.