Class 12 maths NCERT Solutions Exercise 7.1-Integral
Class 12 maths NCERT Solutions Exercise 7.1 of Chapter 7-Integrals-Here NCERT solutions of class 12 maths exercise 7.1 of chapter 12 are the solutions of the chapter 7 exercise 7.1-integrals of class maths NCERT text book . All these NCERT solutions are well explained by the maths expert through a step by step method so that every student could be assessable to the exercise 7.1 of chapter 7-Integral. The purpose of these NCERT Solutions of exercise 7.1 is to clear the concept of integration among the students of class 12.
What is Integrals? Integration is just opposite to differentiation,to understand integration let’s have a function (3x +2), its derivative is 3.Then we can say integral of 3 is 3x +C where C =2 is known as an arbitrary constant. It can be cleared by the following examples.
y = 3x, dy/dx=3
y =3x +2,dy/dx =3
y= 3x +3,dy/dx =3
The differentiation of y with respect to x is the same 3, so we can say that integration of 3 is 3x +C, where C may be any number, further, we can say that 3x +C is the indefinite integration(integral) of 3 , in other words, it can also be called that 3x+C is the anti-derivative of 3.
If there is a function f(x) such that its derivative is g(x),then f(x) is anti-derivative(Integral) of the function g(x).Therefore when the differentiation of a function is integrated we get the original function,in simple words integration is opposite of diffrentiation.
The d/dx(xn) = nxn-1
Here, integral of nxn-1 is xn
Integration is represented by the sign ∫ and dx is attached at the end of the function which means integration of the function with respect to x. The integral of a function is calculated as follows.
Application of integration: When differentiated parts of a region within limits combine together we get the integral of the function within the region. So, integration is used to calculate physical quantities like area, volume, mass, etc.It is widely used in science and technology for the evaluation of the different type of physical quantities and in business predicting the outcomes of particular inputs applied.
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Class 11 NCERT solutions Chapter 1 of maths, physics and chemistry
NCERT Solutions Exercise 7.1-Integral
Q1.Find an anti-derivative (or integral) of the following functions by the method of inspection
We know -sinθ is the derivative of cosθ,so differentiation of cosθ is =-sin
Hence anti-derivative of sin2x is -1/2(cos2x).
Q2.Find an anti-derivative (or integral) of the following functions by the method of inspection
Ans. We know cosθ is the derivative of sinθ
Therefore anti-derivative (integral) of cos3x is 1/3(sin3x).
Q3.Find an anti-derivative (or integral) of the following functions by the method of inspection
Ans.We know the derivative of is ,so
Therefore anti-derivative of is 1/2().
Q4.Find an anti-derivative (or integral) of the following functions by the method of inspection
Ans.We know the derivative of x³ is 3x², so
Therefore antiderivative of (ax+b)² is 1/3a(ax+b)³
Q5.Find an anti-derivative (or integral) of the following functions by the method of inspection
Ans.We know -sinθ is the derivative of cosθ
We know derivative of is
Subtracting equation (ii) from (i), we get
Therefore antiderivative of is
Find the following integrals from Exercises 6 to 20.
Writting the function in expanded form
Dividing numerator by the denominator, we get x² +1
Ans. On multiplication, we get,
By taking separately, we get,
= tan x + sec x + C
Ans. We get,
On further calculation, we get,
By taking seriously, we get
= tan x – x + C
Therefore the correct answer is (C)
Ans. We are given
Therefore anti-derivative of is f(x)
We are given f(2) =0
Substituting this value of C in f(x)
Therefore the correct answer is (A)
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