MCQ
$\mathop \smallint \limits_2^4 \frac{{\log {x^2}}}{{\log {x^2} + {\rm{log}}\left( {36 - 12x + {x^2}} \right)}}\;dx = $
- A$6$
- B$2$
- C$4$
- ✓$1$
$\int\limits_a^b {f\left( x \right)} dx = \int\limits_a^b {f\left( {a + b - x} \right)} dx$ and then add
Let $I = \int\limits_2^4 {\frac{{\log {x^2}}}{{\log {x^2} + \log \left( {36 - 12x + {x^2}} \right)}}} dx$
$I = \int\limits_2^4 {\frac{{2\log xdx}}{{2\log {x^2} + \log {{\left( {6 - x} \right)}^2}}}} dx$
$\int\limits_2^4 {\frac{{2\log x}}{{2[\log x + \log (6 - x)]}}} $
$ \Rightarrow I = \int\limits_2^4 {\frac{{\log xdx}}{{\left| {\log x + \log \left( {6 - x} \right)} \right|}}} ...........\left( 1 \right)$
$ \Rightarrow I = \int\limits_2^4 {\frac{{\log \left( {6 - x} \right)}}{{\log \left( {6 - x} \right) + \log x}}dx} ...........\left( 2 \right)$
$\left[ {\therefore \int\limits_a^b {f\left( x \right)} dx = \int\limits_a^b {f\left( {a + b - x} \right)} dx} \right]$
On adding Eqs. $(i)$ and $(ii),$ we get
$2I = \int\limits_2^4 {\frac{{\log x + \log \left( {6 - x} \right)}}{{\log x + \log \left( {6 - x} \right)}}dx} $
$ \Rightarrow 2I = \int\limits_2^4 {dx\left[ x \right]_2^4} $
$ \Rightarrow 2I = 2$
$\Rightarrow \quad 2 I=2 \Rightarrow I=1$
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
$($ where $det(B)$ denotes determinant of Matrix $B) -$
$I=\int_{0}^{10} \frac{[x] e^{[x]}}{e^{x-1}} d x,$
where $[ x ]$ denotes the greatest integer less than or equal to $x$. Then the value of $I$ is equal to:
$x+(\cos \gamma) y+(\cos \beta) z=0$
$(\cos \gamma) x+y+(\cos \alpha) z=0$
$(\cos \beta) x+(\cos \alpha) y+z=0$
has :
$\frac{1}{2}$
$\frac{1}{3}$
$\frac{1}{4}$
$\text{None of these}$