Question

How do you integrate $ \dfrac{{x - 2}}{{x - 1}} $ ?

Answer 1

Hint: In order to determine the answer of the above indefinite integral , split the numerator as $ x - 1 - 1 $ and separate the denominator into two terms. Using the rule of integration that the integration of one is equal to x and integration of $ \dfrac{1}{{ax + c}} $ is equal to $ \ln \left| {ax + b} \right| + C $ to get your required result.
Formula:
 $
  \int {{x^n}dx = \dfrac{{{x^{n + 1}}}}{{n + 1}} + C} \\
  \int {\dfrac{1}{{ax + b}}} = \ln \left| {ax + b} \right| + C \\
  $

Complete step by step solution:
We are given a expression $ \dfrac{{x - 2}}{{x - 1}} $ ---------(1)
 $ I = \int {\dfrac{{x - 2}}{{x - 1}}dx} $
Let’s rewrite the numerator as $ x - 1 - 1 $ ,we get
 $ I = \int {\dfrac{{x - 1 - 1}}{{x - 1}}dx} $
Now separating the terms , our equation becomes
 $
  I = \int {\dfrac{{x - 1}}{{x - 1}} - \dfrac{1}{{x - 1}}dx} \\
   = \int {1 - \dfrac{1}{{x - 1}}dx} \;
  $
AS we know integration of one is equal to $ x $ and integration of $ \dfrac{1}{{ax + c}} $ is equal to $ \ln \left| {ax + b} \right| + C $
 $ = x - \ln \left| {x - 1} \right| + C $ where C is the constant of integration.
Therefore, the integration of the expression $ \dfrac{{x - 2}}{{x - 1}} $ is equal to $ x - \ln \left| {x - 1} \right| + C $ where C is the Constant of integration.
So, the correct answer is “ $ x - \ln \left| {x - 1} \right| + C $ ”.

Note: 1.Different types of methods of Integration:
Integration by Substitution
Integration by parts
Integration of rational algebraic function by using partial fraction
2. Integration by Substitution :The method of evaluating the integral by reducing it to standard form by a proper substitution is called integration by substitution .
3. Constant of Integration is always placed after the integration. Constant integration gives the family of functions. It allows us to give the anti-derivatives in general form.