( x - 1) 1 + ( x) ^2 ^2 dx is equal to - Vidyadip

MCQ

${\int {\left\{ {\frac{{(\log x - 1)}}{{1 + {{(\log x)}^2}}}} \right\}} ^2}dx$ is equal to

A
$\frac{{x{e^x}}}{{1 + {x^2}}} + c$
✓
$\frac{x}{{{{(\log x)}^2} + 1}} + C$
C
$\frac{{\log x}}{{{{(\log x)}^2} + 1}} + c$
D
$\frac{x}{{{x^2} + 1}} + c$

✓

Answer

Correct option: B.

$\frac{x}{{{{(\log x)}^2} + 1}} + C$

b
(b) ${\int {\left\{ {\frac{{\log x - 1}}{{1 + {{(\log x)}^2}}}} \right\}} ^2}dx$.

Put $\log x = t \Rightarrow dx = {e^t}dt$
Integral $ = \int {{e^t}\left[ {\frac{1}{{1 + {t^2}}} - \frac{{2t}}{{{{(1 + {t^2})}^2}}}} \right]} \;dt$
$\left[ {\because \;\int {{e^x}[f(x) + f'(x)]\;dx = {e^x}f(x) + c} \;} \right]$
$ = \frac{{{e^t}}}{{1 + {t^2}}} + C = \frac{x}{{1 + {{(\log x)}^2}}} + C$.

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "M.C.Q (1 Marks)" from 7.1 inderfinite integral→7.1 inderfinite integral — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Product of all the solution of the equation ${x^{1 + {{\log }_{10}}x}} = 100000x$ is

The rate of growth of bacteria in a culture is proportional to the number of bacteris present and the bacteria count is $1000$ at initial time $t =0 .$ The number of bacteria is increased by $20 \%$ in $2$ hours. If the population of bacteria is $2000$ after $\frac{ k }{\log _{ e }\left(\frac{6}{5}\right)}$ hours, then $\left(\frac{ k }{\log _{ c } 2}\right)^{2}$ is equal to

The value of $\sum\limits_{n = 1}^N {{U_n},} $ if ${U_n} = \left| {\,\begin{array}{*{20}{c}}n&1&5\\{{n^2}}&{2N + 1}&{2N + 1}\\{{n^3}}&{3{N^2}}&{3N}\end{array}\,} \right|$ is

$\int \limits_0^{\infty} \frac{6}{e^{3 x}+6 e^{2 x}+11 e^x+6} d x$

If $A=\left[\begin{array}{rrr}1 & -2 & 1 \\ 2 & 1 & 3\end{array}\right]$ and $B=\left[\begin{array}{ll}2 & 1 \\ 3 & 2 \\ 1 & 1\end{array}\right]$, then $(A B)^T$ is equal to

Let $2\hat a = \hat b \times \hat c + 2\hat b$ then sum of possible value$(s)$ .of $\left| {2\hat a + \hat b + \hat c} \right|$ is

Number of solution(s) satisfying the equation, $3x^2 - 2x^3 = log_2 (x^2 + 1) -log_2 x$ is

The minimum value of $4{e^{2x}} + 9{e^{ - 2x}}$ is

If ${I_n} = {{{d^n}} \over {d{x^n}}}({x^n}\log x),$ then ${I_n} - n{I_{n - 1}} = $

If $\sin^{-1}\text{x}-\cos^{-1}\text{x}=\frac{\pi}{6},$ then x =

$\frac{1}{2}$
$\frac{\sqrt3}{2}$
$-\frac{1}{2}$
$\text{none of these}$