Mathematics

Solve: $$\int {\dfrac{{{\mathop{\rm logx}\nolimits} }}{{{{\left( {1 + \log x} \right)}^2}}}dx} $$


SOLUTION
$$\begin{array}{l}\int {\cfrac{{{\mathop{\rm logx}\nolimits} }}{{{{\left( {1 + \log x} \right)}^2}}}dx} \\Let,\log x = t\\x = {e^t}\\dx = xdt\\\therefore I = \int {\cfrac{t}{{{{\left( {1 + t} \right)}^2}}}xdt}  = \int {\cfrac{{{e^t}.t}}{{{{\left( {1 + t} \right)}^2}}}dt}  = \int {\cfrac{{{e^t}.\left( {t + 1 - 1} \right)}}{{{{\left( {1 + t} \right)}^2}}}dt} \\ = \int {\cfrac{{{e^t}.\left( {t + 1} \right)}}{{{{\left( {1 + t} \right)}^2}}}dt}  - \int {\cfrac{{{e^t}}}{{\left( {1 + {t^2}} \right)}}dt} \\ = \int {{e^t}\left[ {\cfrac{1}{{1 + t}} - \cfrac{1}{{1 + {t^2}}}} \right]} dx\\ = \int {{e^t}\left[ {f(x) + f'(x)} \right]} x\\ = {e^t}f(x) + C\\ = {e^t}\cfrac{1}{{1 + t}} + C\\ = \cfrac{x}{{1 + \log x}} + C\end{array}$$
View Full Answer

Its FREE, you're just one step away


Subjective Medium Published on 17th 09, 2020
Next Question
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 84
Enroll Now For FREE

Realted Questions

Q1 Single Correct Medium
Evaluate $$\displaystyle \int_{0}^{\pi /2} \frac{dx}{2+\sin 2x}$$
  • A. $$\displaystyle \frac{\pi }{{3}}$$
  • B. $$\displaystyle \frac{2\pi }{{5}}$$
  • C. None of these
  • D. $$\displaystyle \frac{2\pi }{{3}}$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q2 Subjective Medium
Integrate
$$(4x +2)\sqrt{x^{2}+x+1}   dx$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q3 Single Correct Medium
$$\int e^{x}[\frac{x^{2}+1}{(x+1)^{2}}]dx$$=
  • A. $$e^{x}(\frac{x+1}{x-1})+c$$
  • B. $$e^{x}\frac{x-1}{(x+1)^{2}}+c$$
  • C. $$e^{x}\frac{x+1}{(x-1)^{2}}+c$$
  • D. $$e^{x}(\frac{x-1}{x+1})+c$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q4 Single Correct Medium
$$\displaystyle \int _{ 0 }^{ a }{ \frac { dx }{ a+\sqrt { { a }^{ 2 }-{ x }^{ 2 } }  }  } $$ is equal to
  • A. $$\displaystyle \frac { \pi  }{ 2 } +1$$
  • B. $$\displaystyle 1-\frac { \pi  }{ 2 } $$
  • C. none of these
  • D. $$\displaystyle \frac { \pi  }{ 2 } -1$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q5 Passage Hard
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 $$(0, 1)$$
Then answer the following question.

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer