1
JEE Main 2023 (Online) 25th January Evening Shift
Numerical
+4
-1

If $$\int\limits_{{1 \over 3}}^3 {|{{\log }_e}x|dx = {m \over n}{{\log }_e}\left( {{{{n^2}} \over e}} \right)}$$, where m and n are coprime natural numbers, then $${m^2} + {n^2} - 5$$ is equal to _____________.

2
JEE Main 2023 (Online) 24th January Evening Shift
Numerical
+4
-1

Let $$f$$ be $$a$$ differentiable function defined on $$\left[ {0,{\pi \over 2}} \right]$$ such that $$f(x) > 0$$ and $$f(x) + \int_0^x {f(t)\sqrt {1 - {{({{\log }_e}f(t))}^2}} dt = e,\forall x \in \left[ {0,{\pi \over 2}} \right]}$$. Then $$\left( {6{{\log }_e}f\left( {{\pi \over 6}} \right)} \right)^2$$ is equal to __________.

3
JEE Main 2023 (Online) 24th January Morning Shift
Numerical
+4
-1

The value of $$12\int\limits_0^3 {\left| {{x^2} - 3x + 2} \right|dx}$$ is ____________

4
JEE Main 2023 (Online) 24th January Morning Shift
Numerical
+4
-1

The value of $${8 \over \pi }\int\limits_0^{{\pi \over 2}} {{{{{(\cos x)}^{2023}}} \over {{{(\sin x)}^{2023}} + {{(\cos x)}^{2023}}}}dx}$$ is ___________