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
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
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
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