Class 12 Maths well explained NCERT solutions of exercise 7.2 of chapter 7-Integrals - Future Study Point

Class 12 Maths well explained NCERT solutions of exercise 7.2 of chapter 7-Integrals

Class 12 Maths well explained NCERT solutions of exercise 7.2 of chapter 7-Integrals

Class 12 Maths well explained NCERT solutions of exercise 7.2 of chapter 7-Integrals

Here Class 12 Maths well explained NCERT solutions of exercise 7.2 of chapter 7-Integrals are created to clear your concept of methods solving integration of a function. These NCERT  solutions are the solutions of exercise 7.1 of chapter 7-Integrals of Class 12 maths NCERT textbook prescribed by CBSE. All questions of exercise 7.2 of chapter 7-Integrals are solved by the expert by a step by step method. Although there are several methods used for solving integration but the method used in the exercise 7.2 for solving the integral functions is substitution method.

Class 12 Maths well explained NCERT solutions of exercise 7.2 of chapter 7-Integrals

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What is the substitution method for calculation of integral function?-Let two functions exist in the way ∫f(x).f'(x) dx, here second function is the derivative of the first function.

Let t = f(x)

Differentiating it both sides

dt/dx = f'(x)

dt = f'(x) dx

Substituting it in the integral

∫t dt = t²/2 +C , now substituting t = f(x) back in the integration

∫f(x).f'(x) dx = [f(x)]²/2

As an example Let’s calculate ∫2x cosx² dx =?

We have to find out

∫2x cosx² dx

here d/dx of x² is 2x, so ,

Let t = x²

dt/dx = 2x⇒ dt = 2x.dx

Substituting t = x² and dt = 2x dx in the integration, we have

∫cost dt = sint + C

Now substituting t = x² back to the integrals

∫cost dt = sin x² + C

Now, you can easily understand the method used here for calculating integration of the function used here in exercise 7.2 of chapter 7-Integrals

Class 12 Maths NCERT solutions of Chapter 7 Integrals

Exercise 7.1- Integrals

Exercise 7.2-Integral

Exercise 7.3-Integrals

Exercise 7.4 -Integral

 

NCERT solutions of exercise 7.2 of chapter 7-Integrals

Integrate the function in exercises 1 to 37.

Let 1 + x² = t

dt/dx = 2x

dt = 2x dx

We have

Ans.

Since 1/x is the d/dx of    and

So, let t =

Substituting the value of dt

Substituting  t =     back in the integral

Hence the integration of the given function is   

Ans. The given function is

Integrating it

Let 1 + logx =t

dt/dx = 1/x

dt =dx/x

Substituting the value of dt

Substituting the value of t back in the integral

Hence the required integration of the given function is 

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Q4. sinx.sin(cosx)

Ans. We are given sinx.sin(cosx)

Integrating it

Since d/dx of cosx =-sinx, so

Let t = cosx

dt/dx = -sinx

dt = -sinx dx

sinxdx= -dt

Substituting the value of sinx dx

= -(-cost) + C

= cost + C

Substituting back the value of t= cosx

= cos (cosx) + C

Q5. sin(ax +b) cos(ax +b)

Ans. We are given the function

sin(ax +b) cos(ax +b)

Divuding and multipling  it by 2, we have

Now,integrating it

Let 2(ax+b) = t

dt/dx = 2a

dt/2a =  dx

Substituting the value of dx

 

Subtituting back the value of t

 

Ans. We are given

Integrating it

Let t = ax +b

dt/dx = a

dx = dt/a

Substituting the value of dx and (ax + b)

Ans. We are given the function

Integrating it

Let t = x + 2⇒ x = t -2

dt/dx = 1⇒ dt = dx

Substituting value of x and dx

Substituting back the value of t

Therefore required integration of the given function is

 

Ans. We are given the function

Integrating it

Let t = 1 + 2x²

dt/dx = 4x

xdx = dt/4

Substituting the value of xdx and (1 + 2x²), we have

Substituting the value of t back

Therefore the required value of integration is   

Ans. We are given the function

Rewriting it

Integrating it

Let t = x² + x + 1

dt/dx = 2x + 1

We have dt = (2x +1)dx

Substituting back the value of t

Class 11 NCERT solutions of Physics and Chemistry

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