Mathematics

# $\displaystyle \int {\left\{ {\dfrac{1}{{\log x}} - \dfrac{1}{{{{\left( {\log x} \right)}^2}}}} \right\}dx}$

##### SOLUTION
$\int \left\{\dfrac{1}{\log\,x}-\dfrac{1}{(\log\,x)^2}\right\}dx$
Now, take $\log\,x=t$
$\Rightarrow$  $x=e^t$
$\Rightarrow$  $dx=e^tdt$
The given integral becomes,
$\int\left(\dfrac{1}{t}\right)-\left(\dfrac{1}{t^2}\right)dt$
$\Rightarrow$  $\int e^t\left(\dfrac{1}{t}-\dfrac{1}{t^2}\right)dt$
Now, we have a formula,
$\int e^x(f(x)+f'(x))dx=e^xf(x)$         ------ ( 1 )
So here, left $f(x)=\dfrac{1}{t}\Rightarrow \,\,f'(x)=\dfrac{-1}{t^2}$
So, it is of form ( 1 )
$\therefore$  The given integral $=e^xf(x)=e^t(1/t)+c$
Substituting value of $t$ we get,
$\Rightarrow$  $e^{\log\,x}(\dfrac{1}{\log\,x})+c$
$\Rightarrow$  $\dfrac{x}{\log\,x}+c$

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Subjective Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 86

#### Realted Questions

Q1 Single Correct Hard
$\displaystyle\int { \cfrac { \sqrt { x } }{ \sqrt { x } -\sqrt [ 3 ]{ x } } } dx$ is equal to
• A. $6\left\{ \cfrac { x }{ 6 }+ \cfrac { { x }^{ 6/5 } }{ 5 } +\cfrac { { x }^{ 1/2 } }{ 2 } +\cfrac { { x }^{ 1/3 } }{ 3 } +\log { \left( { x }^{ 1/6 }-1 \right) } \right\} +c$
• B. $6\left\{ \cfrac { x }{ 6 }+ \cfrac { { x }^{ 6/5 } }{ 5 } +\cfrac { { x }^{ 1/2 } }{ 3 } +\cfrac { { x }^{ 1/3 } }{ 2 } +\log { \left( { x }^{ 1/6 }-1 \right) } \right\} +c$
• C. $6\left\{ \cfrac { x }{ 6 }+ \cfrac { { x }^{ 6/5 } }{ 5 } +\cfrac { { x }^{ 1/2 } }{ 2 } +\cfrac { { x }^{ 1/3 } }{ 3 } +{ x }^{ 1/6 }+\log { \left( { x }^{ 1/6 }-1 \right) } \right\} +c$
• D. None of the above

1 Verified Answer | Published on 17th 09, 2020

Q2 Assertion & Reason Hard
##### ASSERTION

If $l_n=\int \cot^n x dx$ then $5(I_6+I_4)=-\cot x$

##### REASON

If $l_n=\int \cot^n x dx$ then $l_n\displaystyle = -\frac{\cot^{n-1} x}{n}-I_{n-2}$ where $n\geq 2$

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Both Assertion and Reason are incorrect

1 Verified Answer | Published on 17th 09, 2020

Q3 Subjective Hard
Prove that $\displaystyle \int_{0}^{{\pi}/{4}} (\sqrt {\tan x} + \sqrt {\cot x})dx = \sqrt {2}\cdot \dfrac {\pi}{2}$

1 Verified Answer | Published on 17th 09, 2020

Q4 Single Correct Medium
The value of $\displaystyle\int\limits_{-1}^1x|x|\ dx=$
• A. $\dfrac{1}{2}$
• B. $1$
• C. none of these
• D. $0$

Given that for each $\displaystyle a \in (0, 1), \lim_{h \rightarrow 0^+} \int_h^{1-h} t^{-a} (1 -t)^{a-1}dt$ exists. Let this limit be $g(a)$
In addition, it is given that the function $g(a)$ is differentiable on $(0, 1)$