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.