1
JEE Main 2022 (Online) 27th June Morning Shift
+4
-1

The value of the integral

$$\int\limits_{ - 2}^2 {{{|{x^3} + x|} \over {({e^{x|x|}} + 1)}}dx}$$ is equal to :

A
5e2
B
3e$$-$$2
C
4
D
6
2
JEE Main 2022 (Online) 25th June Evening Shift
+4
-1

If $${b_n} = \int_0^{{\pi \over 2}} {{{{{\cos }^2}nx} \over {\sin x}}dx,\,n \in N}$$, then

A
$${b_3} - {b_2},\,{b_4} - {b_3},\,{b_5} - {b_4}$$ are in A.P. with common difference $$-$$2
B
$${1 \over {{b_3} - {b_2}}},{1 \over {{b_4} - {b_3}}},{1 \over {{b_5} - {b_4}}}$$ are in an A.P. with common difference 2
C
$${b_3} - {b_2},\,{b_4} - {b_3},\,{b_5} - {b_4}$$ are in a G.P.
D
$${1 \over {{b_3} - {b_2}}},{1 \over {{b_4} - {b_3}}},{1 \over {{b_5} - {b_4}}}$$ are in an A.P. with common difference $$-$$2
3
JEE Main 2022 (Online) 25th June Morning Shift
+4
-1

The value of $$\int\limits_0^\pi {{{{e^{\cos x}}\sin x} \over {(1 + {{\cos }^2}x)({e^{\cos x}} + {e^{ - \cos x}})}}dx}$$ is equal to:

A
$${{{\pi ^2}} \over 4}$$
B
$${{{\pi ^2}} \over 2}$$
C
$${\pi \over 4}$$
D
$${\pi \over 2}$$
4
JEE Main 2022 (Online) 24th June Evening Shift
+4
-1

The value of the integral

$$\int\limits_{ - \pi /2}^{\pi /2} {{{dx} \over {(1 + {e^x})({{\sin }^6}x + {{\cos }^6}x)}}}$$ is equal to

A
2$$\pi$$
B
0
C
$$\pi$$
D
$${\pi \over 2}$$
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