1
JEE Main 2021 (Online) 27th August Evening Shift
+4
-1
The value of the integral $$\int\limits_0^1 {{{\sqrt x dx} \over {(1 + x)(1 + 3x)(3 + x)}}}$$ is :
A
$${\pi \over 8}\left( {1 - {{\sqrt 3 } \over 2}} \right)$$
B
$${\pi \over 4}\left( {1 - {{\sqrt 3 } \over 6}} \right)$$
C
$${\pi \over 8}\left( {1 - {{\sqrt 3 } \over 6}} \right)$$
D
$${\pi \over 4}\left( {1 - {{\sqrt 3 } \over 2}} \right)$$
2
JEE Main 2021 (Online) 27th August Morning Shift
+4
-1
Out of Syllabus
If $${U_n} = \left( {1 + {1 \over {{n^2}}}} \right)\left( {1 + {{{2^2}} \over {{n^2}}}} \right)^2.....\left( {1 + {{{n^2}} \over {{n^2}}}} \right)^n$$, then $$\mathop {\lim }\limits_{n \to \infty } {({U_n})^{{{ - 4} \over {{n^2}}}}}$$ is equal to :
A
$${{{e^2}} \over {16}}$$
B
$${4 \over e}$$
C
$${{16} \over {{e^2}}}$$
D
$${4 \over {{e^2}}}$$
3
JEE Main 2021 (Online) 27th August Morning Shift
+4
-1
$$\int\limits_6^{16} {{{{{\log }_e}{x^2}} \over {{{\log }_e}{x^2} + {{\log }_e}({x^2} - 44x + 484)}}dx}$$ is equal to :
A
6
B
8
C
5
D
10
4
JEE Main 2021 (Online) 26th August Evening Shift
+4
-1
If the value of the integral
$$\int\limits_0^5 {{{x + [x]} \over {{e^{x - [x]}}}}dx = \alpha {e^{ - 1}} + \beta }$$, where $$\alpha$$, $$\beta$$ $$\in$$ R, 5$$\alpha$$ + 6$$\beta$$ = 0, and [x] denotes the greatest integer less than or equal to x; then the value of ($$\alpha$$ + $$\beta$$)2 is equal to :
A
100
B
25
C
16
D
36
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