MCQ
જો $[x]$ એ મહતમ પૃણાંક વિધેય છે તો $\int\limits_{ - 0.9}^{0.9} {\left( {\left[ {{x^2}} \right] + \log \left( {\frac{{2 - x}}{{2 + x}}} \right)} \right)} dx$ ની કિમંત મેળવો.
- A$0.486$
- B$0.243$
- C$1.8$
- ✓$0$
$ = \int\limits_{ - 0.9}^{0.9} {\left[ {{x^2}} \right]} dx + \int\limits_{ - 0.9}^{0.9} {\log } \left( {\frac{{2 - x}}{{2 + x}}} \right)dx$
$ = 0 + \int\limits_{ - 0.9}^{0.9} {\log \left( {\frac{{2 - x}}{{2 + x}}} \right)} dx$
Put $x = - x \Rightarrow f(x) = \log \frac{{2 - x}}{{2 + x}}$
and $f(-x)=\log \frac{2+x}{2-x}$
$=-\log \frac{(2-x)}{2+x}=-f(x)$
So, it is an odd function, hence Required integral $=0$.
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વિધાન $-2$ : $f(x) = \frac{1}{{\sqrt {1 - {x^2}} }} + \left[ {\frac{{{x^2} + x + 1}}{4}} \right]$ , જ્યા $[.]$ એ મહત્તમ વિધેય છે. વિધેય $f(x)$ એ યુગ્મ વિધેય છે.