MCQ
જો $\int_0^1 {x\log \left( {1 + \frac{x}{2}} \right)} \,dx = a + b\log \frac{2}{3},$ તો
- A$a = \frac{3}{2},\,\,\,b = \frac{3}{2}$
- B$a = \frac{3}{4},\,\,\,b = - \frac{3}{4}$
- ✓$a = \frac{3}{4},\,\,\,b = \frac{3}{2}$
- D$a = b$
$ = \left[ {\log \left( {1 + \frac{x}{2}} \right)\frac{{{x^2}}}{2}} \right]_0^2 - \int_0^1 {\frac{1}{{1 + \frac{x}{2}}}\frac{1}{2}\frac{{{x^2}}}{2}} dx$
$ = \frac{1}{2}\log \frac{3}{2} - \frac{1}{2}\int_0^1 {\frac{{{x^2}}}{{x + 2}}dx} $
$ = \frac{1}{2}\log \frac{3}{2} - \frac{1}{2}\left[ {\frac{1}{2} - 2 + 4\log 3 - 4\log 2} \right] $
$= \frac{3}{4} + \frac{3}{2}\log \frac{2}{3}$
On comparing with the given value $a = \frac{3}{4},b = \frac{3}{2}$.
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.