1
JEE Main 2019 (Online) 11th January Evening Slot
+4
-1
The integral  $$\int\limits_{\pi /6}^{\pi /4} {{{dx} \over {\sin 2x\left( {{{\tan }^5}x + {{\cot }^5}x} \right)}}}$$  equals :
A
$${\pi \over {40}}$$
B
$${1 \over {20}}{\tan ^{ - 1}}\left( {{1 \over {9\sqrt 3 }}} \right)$$
C
$${1 \over {10}}\left( {{\pi \over 4} - {{\tan }^{ - 1}}\left( {{1 \over {9\sqrt 3 }}} \right)} \right)$$
D
$${1 \over 5}\left( {{\pi \over 4}{{-\tan }^{ - 1}}\left( {{1 \over {3\sqrt 3 }}} \right)} \right)$$
2
JEE Main 2019 (Online) 11th January Morning Slot
+4
-1
The value of the integral $$\int\limits_{ - 2}^2 {{{{{\sin }^2}x} \over { \left[ {{x \over \pi }} \right] + {1 \over 2}}}} \,dx$$ (where [x] denotes the greatest integer less than or equal to x) is
A
0
B
4
C
4$$-$$ sin 4
D
sin 4
3
JEE Main 2019 (Online) 10th January Evening Slot
+4
-1
The value of   $$\int\limits_{ - \pi /2}^{\pi /2} {{{dx} \over {\left[ x \right] + \left[ {\sin x} \right] + 4}}} ,$$  where [t] denotes the greatest integer less than or equal to t, is
A
$${1 \over {12}}\left( {7\pi - 5} \right)$$
B
$${1 \over {12}}\left( {7\pi + 5} \right)$$
C
$${3 \over {10}}\left( {4\pi - 3} \right)$$
D
$${3 \over {20}}\left( {4\pi - 3} \right)$$
4
JEE Main 2019 (Online) 10th January Evening Slot
+4
-1
If  $$\int\limits_0^x \,$$f(t) dt = x2 + $$\int\limits_x^1 \,$$ t2f(t) dt then f '$$\left( {{1 \over 2}} \right)$$ is -
A
$${{18} \over {25}}$$
B
$${{6} \over {25}}$$
C
$${{24} \over {25}}$$
D
$${{4} \over {5}}$$
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