1
JEE Main 2021 (Online) 26th August Evening Shift
MCQ (Single Correct Answer)
+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
2
JEE Main 2021 (Online) 26th August Evening Shift
MCQ (Single Correct Answer)
+4
-1
The value of $$\int\limits_{ - {\pi \over 2}}^{{\pi \over 2}} {\left( {{{1 + {{\sin }^2}x} \over {1 + {\pi ^{\sin x}}}}} \right)} \,dx$$ is
A
$${\pi \over 2}$$
B
$${{5\pi } \over 4}$$
C
$${{3\pi } \over 4}$$
D
$${{3\pi } \over 2}$$
3
JEE Main 2021 (Online) 26th August Morning Shift
MCQ (Single Correct Answer)
+4
-1
The value of $$\int\limits_{{{ - 1} \over {\sqrt 2 }}}^{{1 \over {\sqrt 2 }}} {{{\left( {{{\left( {{{x + 1} \over {x - 1}}} \right)}^2} + {{\left( {{{x - 1} \over {x + 1}}} \right)}^2} - 2} \right)}^{{1 \over 2}}}dx}$$ is :
A
loge 4
B
loge 16
C
2loge 16
D
4loge (3 + 2$${\sqrt 2 }$$)
4
JEE Main 2021 (Online) 26th August Morning Shift
MCQ (Single Correct Answer)
+4
-1
Out of Syllabus
The value of

$$\mathop {\lim }\limits_{n \to \infty } {1 \over n}\sum\limits_{r = 0}^{2n - 1} {{{{n^2}} \over {{n^2} + 4{r^2}}}}$$ is :
A
$${1 \over 2}{\tan ^{ - 1}}(2)$$
B
$${1 \over 2}{\tan ^{ - 1}}(4)$$
C
$${\tan ^{ - 1}}(4)$$
D
$${1 \over 4}{\tan ^{ - 1}}(4)$$
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