   Question Answers

# Find the integration of $\int {\dfrac{{1 - {x^2}}}{{x\left( {1 - 2x} \right)}}dx}$.  Hint: In this question first of all multiply and divide with 2 both in numerator and denominator to break the integration into a simplified form. Use the concept of partial fractions to resolve the integration further to simply it.

Let $I = \int {\dfrac{{1 - {x^2}}}{{x\left( {1 - 2x} \right)}}dx}$
Multiply and divide by 2 in the integral we get,
$I = \int {\dfrac{{2 - 2{x^2}}}{{2x\left( {1 - 2x} \right)}}dx} = \int {\dfrac{{2{x^2} - 2}}{{2x\left( {2x - 1} \right)}}dx}$
Now the above integral is also written as
$\Rightarrow I = \int {\left( {\dfrac{{x - 2}}{{2x\left( {2x - 1} \right)}} + \dfrac{1}{2}} \right)dx}$, $\left[ {\because \dfrac{{2{x^2} - 2}}{{2x\left( {2x - 1} \right)}} = \dfrac{{x - 2}}{{2x\left( {2x - 1} \right)}} + \dfrac{1}{2}} \right]$
Apply partial fraction
So let,
$\dfrac{{x - 2}}{{2x\left( {2x - 1} \right)}} = \dfrac{A}{{2x}} + \dfrac{B}{{2x - 1}}$
$\Rightarrow x - 2 = A\left( {2x - 1} \right) + B\left( {2x} \right)$
Now comparing the terms we have,
$\Rightarrow 2A + 2B = 1$……………………… (1) (Comparing x terms)
And $- A = - 2 \Rightarrow A = 2$
So from equation (1) we have,
$\begin{gathered} \Rightarrow 2\left( 2 \right) + 2B = 1 \\ \Rightarrow 2B = 1 - 4 = - 3 \\ \Rightarrow B = \dfrac{{ - 3}}{2} \\ \end{gathered}$
$\Rightarrow \dfrac{{x - 2}}{{2x\left( {2x - 1} \right)}} = \dfrac{2}{{2x}} + \dfrac{{\dfrac{{ - 3}}{2}}}{{2x - 1}}$
So the integral becomes
$\Rightarrow I = \int {\left( {\dfrac{2}{{2x}} + \dfrac{{\dfrac{{ - 3}}{2}}}{{2x - 1}} + \dfrac{1}{2}} \right)dx}$
$\Rightarrow I = \int {\left( {\dfrac{1}{x} - \dfrac{3}{{2\left( {2x - 1} \right)}} + \dfrac{1}{2}} \right)dx}$
Now integrate the above equation we have, as we know integration of $\dfrac{1}{x}$ is ln x and integration of $\dfrac{1}{{ax + b}}$ is $\dfrac{{\ln \left( {ax + b} \right)}}{a}$ so use these properties we have,
$\Rightarrow I = \ln x - \dfrac{3}{2}\left( {\dfrac{{\ln \left( {2x - 1} \right)}}{2}} \right) + \dfrac{1}{2}\left( x \right) + c$, where c is some arbitrary integration constant.
So this is the required value of the integration.

Note: Whenever we face such types of problems the key concept is to have a good grasp over the standard integration form just like integration of $\dfrac{1}{x}$ is ln x. This always helps after simplification of the integral to get the right answer.
Integration Formula  Integration by Parts Rule  Differentiation and Integration Formula  Integration By Parts Formula  To Find the Weight of a Given Body Using Parallelogram Law of Vectors  CBSE Class 12 Maths Chapter-1 Relations and Functions Formula  CBSE Class 12 Maths Chapter-8 Application of Integrals Formula  CBSE Class 12 Maths Formulas  CBSE Class 12 Maths Chapter-6 Application of Derivatives Formula  IMO Sample Papers for Class 12  Important Questions for CBSE Class 11 Biology Chapter 22 - Chemical Coordination and integration  Important Questions for CBSE Class 12 Maths Chapter 1 - Relations and Functions  Important Questions for CBSE Class 12 Chemistry Chapter 1 - The Solid State  NCERT Books Free Download for Class 12 Maths Chapter-1 Relations and Functions  Important Questions for CBSE Class 11 English Snapshots Chapter 1 - The Summer of the Beautiful White Horse  Important Questions for CBSE Class 12 Biology Chapter 1 - Reproduction in Organism  Important Questions for CBSE Class 12 Physics Chapter 1 - Electric Charges and Fields  Important Questions for CBSE Class 12 Maths Chapter 8 - Application of Integrals  Important Questions for CBSE Class 12 Maths Chapter 6 - Application of Derivatives  CBSE Class 8 Science Reaching The Age of Adolescence Worksheets  Maths Question Paper for CBSE Class 12 - 2016 Set 1 E  Maths Question Paper for CBSE Class 12 - 2016 Set 1 S  Maths Question Paper for CBSE Class 12 - 2016 Set 1 C  Maths Question Paper for CBSE Class 12 - 2016 Set 1 N  CBSE Class 12 Maths Question Paper 2020  Maths Question Paper for CBSE Class 12 - 2013  Chemistry Question Paper for CBSE Class 12 - 2016 Set 1 E  Chemistry Question Paper for CBSE Class 12 - 2016 Set 1 S  Chemistry Question Paper for CBSE Class 12 - 2016 Set 1 N  Chemistry Question Paper for CBSE Class 12 - 2016 Set 1 C  RS Aggarwal Class 12 Solutions Chapter-13 Method of Integration  NCERT Solutions for Class 11 Biology Chapter 22  RS Aggarwal Class 12 Solutions Chapter-15 Integration Using Partial Fractions  RD Sharma Class 12 Solutions Chapter 1 - Relations (Ex 1.1) Exercise 1.1  NCERT Solutions for Class 12 Maths Chapter 1 Exercise 1.1 (Ex 1.1)  RD Sharma Class 12 Solutions Chapter 1 - Relations (Ex 1.2) Exercise 1.2  RD Sharma Class 12 Solutions Chapter 12 - Higher Order Derivatives (Ex 12.1) Exercise 12.1  NCERT Solutions for Class 12 Maths Chapter 1 Exercise 1.3 (Ex 1.3)  NCERT Solutions for Class 12 Maths Chapter 1 Exercise 1.4 (Ex 1.4)  NCERT Solutions for Class 12 Maths Chapter 12 Linear Programming (Ex 12.1) Exercise 12.1  