1
JEE Main 2021 (Online) 26th February Evening Shift
+4
-1
Let $$f(x) = \int\limits_0^x {{e^t}f(t)dt + {e^x}}$$ be a differentiable function for all x$$\in$$R. Then f(x) equals :
A
$${e^{({e^{x - 1}})}}$$
B
$$2{e^{{e^x}}} - 1$$
C
$$2{e^{{e^x} - 1}} - 1$$
D
$${e^{{e^x}}} - 1$$
2
JEE Main 2021 (Online) 26th February Evening Shift
+4
-1
For x > 0, if $$f(x) = \int\limits_1^x {{{{{\log }_e}t} \over {(1 + t)}}dt}$$, then $$f(e) + f\left( {{1 \over e}} \right)$$ is equal to :
A
$${1 \over 2}$$
B
$$-$$1
C
0
D
1
3
JEE Main 2021 (Online) 26th February Morning Shift
+4
-1
The value of $$\int\limits_{ - \pi /2}^{\pi /2} {{{{{\cos }^2}x} \over {1 + {3^x}}}} dx$$ is :
A
$$2\pi$$
B
$${\pi \over 2}$$
C
$$4\pi$$
D
$${\pi \over 4}$$
4
JEE Main 2021 (Online) 26th February Morning Shift
+4
-1
The value of $$\sum\limits_{n = 1}^{100} {\int\limits_{n - 1}^n {{e^{x - [x]}}dx} }$$, where [ x ] is the greatest integer $$\le$$ x, is :
A
100e
B
100(e $$-$$ 1)
C
100(1 + e)
D
100(1 $$-$$ e)
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