1
JEE Main 2019 (Online) 9th January Evening Slot
+4
-1
If   $$f\left( x \right) = \int {{{5{x^8} + 7{x^6}} \over {{{\left( {{x^2} + 1 + 2{x^7}} \right)}^2}}}} \,dx,\,\left( {x \ge 0} \right),$$

$$f\left( 0 \right) = 0,$$    then the value of $$f(1)$$ is :
A
$$-$$ $${1 \over 2}$$
B
$$-$$ $${1 \over 4}$$
C
$${1 \over 2}$$
D
$${1 \over 4}$$
2
JEE Main 2019 (Online) 9th January Morning Slot
+4
-1
For x2 $$\ne$$ n$$\pi$$ + 1, n $$\in$$ N (the set of natural numbers), the integral

$$\int {x\sqrt {{{2\sin ({x^2} - 1) - \sin 2({x^2} - 1)} \over {2\sin ({x^2} - 1) + \sin 2({x^2} - 1)}}} dx}$$ is equal to :

(where c is a constant of integration)
A
$${\log _e}\left| {{1 \over 2}{{\sec }^2}\left( {{x^2} - 1} \right)} \right| + c$$
B
$${1 \over 2}{\log _e}\left| {\sec \left( {{x^2} - 1} \right)} \right| + c$$
C
$${1 \over 2}{\log _e}\left| {{{\sec }^2}\left( {{{{x^2} - 1} \over 2}} \right)} \right| + c$$
D
$${\log _e}\left| {\sec \left( {{{{x^2} - 1} \over 2}} \right)} \right| + c$$
3
JEE Main 2018 (Online) 16th April Morning Slot
+4
-1
If $$\int {{{\tan x} \over {1 + \tan x + {{\tan }^2}x}}dx = x - {K \over {\sqrt A }}{{\tan }^{ - 1}}}$$ $$\left( {{{K\,\tan x + 1} \over {\sqrt A }}} \right) + C,(C\,\,$$ is a constant of integration) then the ordered pair (K, A) is equal to :
A
(2, 1)
B
($$-$$2, 3)
C
(2, 3)
D
($$-$$2, 1)
4
JEE Main 2018 (Offline)
+4
-1
The integral

$$\int {{{{{\sin }^2}x{{\cos }^2}x} \over {{{\left( {{{\sin }^5}x + {{\cos }^3}x{{\sin }^2}x + {{\sin }^3}x{{\cos }^2}x + {{\cos }^5}x} \right)}^2}}}} dx$$

is equal to
A
$${{ - 1} \over {1 + {{\cot }^3}x}} + C$$
B
$${1 \over {3\left( {1 + {{\tan }^3}x} \right)}} + C$$
C
$${{ - 1} \over {3\left( {1 + {{\tan }^3}x} \right)}} + C$$
D
$${1 \over {1 + {{\cot }^3}x}} + C$$
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