Question
Let $${I_1} = \int_0^1 {{e^{ - {x^2}}}} dx,\,{I_2} = \int_0^1 {{e^{ - x}}} {\cos ^2}x\,dx$$ and $${I_3} = \int_0^1 {{e^{ - {x^2}}}{{\cos }^2}x\,} dx.$$ Then :
A.
$${I_1} < {I_2} < {I_3}$$
B.
$${I_3} < {I_2} < {I_1}$$
C.
$${I_2} < {I_1} < {I_3}$$
D.
$${I_2} < {I_3} < {I_1}$$
Answer :
$${I_2} < {I_3} < {I_1}$$
Solution :
$$\eqalign{
& {\text{In }}\left( {0,\,1} \right),\,{e^{ - {x^2}}} > {e^{ - {x^2}}}{\cos ^2}x > 0\,\,\,\,\,\,\,\,\therefore {I_1} > {I_3} \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{e^{ - {x^2}}} > {e^{ - x}} > 0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\therefore {e^{ - {x^2}}}{\cos ^2}x > {e^{ - x}}{\cos ^2}x > 0 \cr
& \therefore {I_3} > {I_2} \cr
& \therefore {I_2} < {I_3} < {I_1} \cr} $$