1
JEE Main 2021 (Online) 25th February Morning Shift
+4
-1
The value of the integral
$$\int {{{\sin \theta .\sin 2\theta ({{\sin }^6}\theta + {{\sin }^4}\theta + {{\sin }^2}\theta )\sqrt {2{{\sin }^4}\theta + 3{{\sin }^2}\theta + 6} } \over {1 - \cos 2\theta }}} \,d\theta$$ is :
A
$${1 \over {18}}{\left[ {9 - 2{{\cos }^6}\theta - 3{{\cos }^4}\theta - 6{{\cos }^2}\theta } \right]^{{3 \over 2}}} + c$$
B
$${1 \over {18}}{\left[ {11 - 18{{\sin }^2}\theta + 9{{\sin }^4}\theta - 2{{\sin }^6}\theta } \right]^{{3 \over 2}}} + c$$
C
$${1 \over {18}}{\left[ {11 - 18{{\cos }^2}\theta + 9{{\cos }^4}\theta - 2{{\cos }^6}\theta } \right]^{{3 \over 2}}} + c$$
D
$${1 \over {18}}{\left[ {9 - 2{{\sin }^6}\theta - 3{{\sin }^4}\theta - 6{{\sin }^2}\theta } \right]^{{3 \over 2}}} + c$$
2
JEE Main 2021 (Online) 24th February Morning Shift
+4
-1
If $$\int {{{\cos x - \sin x} \over {\sqrt {8 - \sin 2x} }}} dx = a{\sin ^{ - 1}}\left( {{{\sin x + \cos x} \over b}} \right) + c$$, where c is a constant of integration, then the ordered pair (a, b) is equal to :
A
(-1, 3)
B
(1, 3)
C
(1, -3)
D
(3, 1)
3
JEE Main 2020 (Online) 5th September Evening Slot
+4
-1
If
$$\int {{{\cos \theta } \over {5 + 7\sin \theta - 2{{\cos }^2}\theta }}} d\theta$$ = A$${\log _e}\left| {B\left( \theta \right)} \right| + C$$,

where C is a constant of integration, then $${{{B\left( \theta \right)} \over A}}$$
can be :
A
$${{2\sin \theta + 1} \over {5\left( {\sin \theta + 3} \right)}}$$
B
$${{2\sin \theta + 1} \over {\sin \theta + 3}}$$
C
$${{5\left( {2\sin \theta + 1} \right)} \over {\sin \theta + 3}}$$
D
$${{5\left( {\sin \theta + 3} \right)} \over {2\sin \theta + 1}}$$
4
JEE Main 2020 (Online) 5th September Morning Slot
+4
-1
If
$$\int {\left( {{e^{2x}} + 2{e^x} - {e^{ - x}} - 1} \right){e^{\left( {{e^x} + {e^{ - x}}} \right)}}dx}$$ = $$g\left( x \right){e^{\left( {{e^x} + {e^{ - x}}} \right)}} + c$$

where c is a constant of integration, then g(0) is equal to :
A
1
B
2
C
e
D
e2
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