1
JEE Main 2017 (Offline)
+4
-1
Let $${I_n} = \int {{{\tan }^n}x\,dx} ,\,\left( {n > 1} \right).$$

If $${I_4} + {I_6}$$ = $$a{\tan ^5}x + b{x^5} + C$$, where C is a constant of integration,

then the ordered pair $$\left( {a,b} \right)$$ is equal to
A
$$\left( {{1 \over 5},0} \right)$$
B
$$\left( {{1 \over 5}, - 1} \right)$$
C
$$\left( { - {1 \over 5},0} \right)$$
D
$$\left( { - {1 \over 5},1} \right)$$
2
JEE Main 2016 (Online) 10th April Morning Slot
+4
-1
The integral $$\int {{{dx} \over {\left( {1 + \sqrt x } \right)\sqrt {x - {x^2}} }}}$$ is equal to :

(where C is a constant of integration.)
A
$$- 2\sqrt {{{1 + \sqrt x } \over {1 - \sqrt x }}} + C$$
B
$$- 2\sqrt {{{1 - \sqrt x } \over {1 + \sqrt x }}} + C$$
C
$$- \sqrt {{{1 - \sqrt x } \over {1 + \sqrt x }}} + C$$
D
$$2\sqrt {{{1 + \sqrt x } \over {1 - \sqrt x }}} + C$$
3
JEE Main 2016 (Online) 9th April Morning Slot
+4
-1
If   $$\int {{{dx} \over {{{\cos }^3}x\sqrt {2\sin 2x} }}} = {\left( {\tan x} \right)^A} + C{\left( {\tan x} \right)^B} + k,$$

where k is a constant of integration, then A + B +C equals :
A
$${{21} \over 5}$$
B
$${{16} \over 5}$$
C
$${{7} \over 10}$$
D
$${{27} \over 10}$$
4
JEE Main 2016 (Offline)
+4
-1
The integral $$\int {{{2{x^{12}} + 5{x^9}} \over {{{\left( {{x^5} + {x^3} + 1} \right)}^3}}}} dx$$ is equal to :
A
$${{{x^5}} \over {2{{\left( {{x^5} + {x^3} + 1} \right)}^2}}} + C$$
B
$${{ - {x^{10}}} \over {2{{\left( {{x^5} + {x^3} + 1} \right)}^2}}} + C$$
C
$${{{-x^5}} \over {{{\left( {{x^5} + {x^3} + 1} \right)}^2}}} + C$$
D
$${{ {x^{10}}} \over {2{{\left( {{x^5} + {x^3} + 1} \right)}^2}}} + C$$
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