   Question Answers

# Find the value of $\int {\dfrac{1}{{x\log x\log \left( {\log x} \right)}}} dx$.A. $\left[ {\log \left( {\log x} \right)} \right]$B. $\log \left[ {\log \left( {\log x} \right)} \right]$C. $\log \left[ {\log \left( {\log {x^2}} \right)} \right]$D. $- \log \left[ {\log \left( {\log x} \right)} \right]$  Hint: This problem can be solved by using a substitution method. According to the substitution method, the given integral can be transformed into another form by changing the independent variable $x{\text{ to }}t$. So, use this concept to reach the solution of the given problem.

Let $I = \int {\dfrac{1}{{x\log x\log \left( {\log x} \right)}}} dx$
Put $\log \left( {\log x} \right) = t$
Differentiating $\log \left( {\log x} \right) = t$ w.r.t $x$, we have
$\Rightarrow \dfrac{d}{{dx}}\left[ {\log \left( {\log x} \right)} \right] = \dfrac{d}{{dx}}\left( t \right) \\ \Rightarrow \dfrac{1}{{\log x}} \times \dfrac{d}{{dx}}\left( {\log x} \right) = \dfrac{{dt}}{{dx}}{\text{ }}\left[ {\because \dfrac{d}{{dx}}\left( {\log x} \right) = \dfrac{1}{x}} \right] \\ \Rightarrow \dfrac{1}{{x\log x}} = \dfrac{{dt}}{{dx}} \\ \therefore dx = x\log xdt....................................................\left( 1 \right) \\$
Using equation (1), we get
$I = \int {\dfrac{1}{{x\log x\left( t \right)}}x\log xdt} \\ I = \int {\dfrac{1}{t}dt} \\ I = \log t + c{\text{ }}\left[ {\because \int {\dfrac{1}{x}dx = \log x + c} } \right] \\$
Substituting $\log \left( {\log x} \right) = t$, we get
$I = \log \left[ {\log \left( {\log x} \right)} \right] + c$
Hence, $I = \int {\dfrac{1}{{x\log x\log \left( {\log x} \right)}}} dx = \log \left[ {\log \left( {\log x} \right)} \right] + c$
Thus, the correct option is B. $\log \left[ {\log \left( {\log x} \right)} \right]$

Note: A constant namely integrating constant that is added to the function obtained by evaluating the indefinite integral of a given function, indicating that all indefinite integrals of the given function differ by, at most, a constant. So, it is necessary to add integrating constants after completion of the integral.

Value of g  Difference Between Log and Ln  Difference Between Left and Right Ventricle  Value of Sin 180  Integration by Parts Rule  To Find the Weight of a Given Body Using Parallelogram Law of Vectors  Difference Between Left Kidney and Right Kidney  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  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  Important Questions for CBSE Class 12 Micro Economics Chapter 4 - Theory of the Firm under Perfect Competition  Important Questions for CBSE Class 12 Chemistry Chapter 1 - The Solid State  Important Questions for CBSE Class 12 Maths Chapter 12 - Linear Programming  Important Questions for CBSE Class 11 English Snapshots Chapter 1 - The Summer of the Beautiful White Horse  Important Questions for CBSE Class 12 Chemistry Chapter 7 - The p-Block Elements  Important Questions for CBSE Class 6 Social Science The Earth Our Habitat Chapter 3 - Motions of the Earth  Important Questions for CBSE Class 9 Science Chapter 4 - Structure of The Atom  CBSE Class 12 Maths Question Paper 2020  Maths Question Paper for CBSE Class 12 - 2013  Previous Year Question Paper for CBSE Class 12 Maths - 2014  CBSE Previous Year Question Papers Class 12 Maths with Solutions  Maths Question Paper for CBSE Class 12 - 2016 Set 1 C  Maths Question Paper for CBSE Class 12 - 2016 Set 1 E  Maths Question Paper for CBSE Class 12 - 2016 Set 1 S  CBSE Class 12 Maths Question Paper 2018 with Solutions - Free PDF  CBSE Class 12 Maths Question Paper 2015 with Solutions - Free PDF  Maths Question Paper for CBSE Class 12 - 2016 Set 1 N  RD Sharma Class 12 Maths Solutions Chapter 15 - Mean Value Theorems  RS Aggarwal Class 12 Solutions Chapter-13 Method of Integration  Textbooks Solutions for CBSE & ICSE Board of Class 6 to 12 Maths & Science  RD Sharma Class 12 Solutions Chapter 15 - Mean Value Theorems (Ex 15.2) Exercise 15.2  RD Sharma Class 12 Solutions Chapter 15 - Mean Value Theorems (Ex 15.1) Exercise 15.1  RD Sharma Class 12 Maths Solutions Chapter 29 - The Plane  NCERT Solutions for Class 12 English Vistas Chapter 3 - Journey to the end of the Earth  NCERT Solutions for Class 12 English Vistas Chapter 6 - On the Face of it  RD Sharma Class 8 Maths Solutions Chapter 13 - Profits, Loss, Discount, and Value Added Tax  RD Sharma Class 12 Maths Solutions Chapter 23 - Algebra of Vectors  