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.

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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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Chapter 5-Complex numbers	Chapter 13- Limits and Derivatives
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