Given,

$${I_1} = \int_0^1 {{e^{ - x}}} {\cos ^2}x{\mkern 1mu} dx;$$

$${I_2} = \int_0^1 {{e^{ - {x^2}}}} {\cos ^2}x{\mkern 1mu} dx$$ and

$${I_3} = \int_0^1 {{e^{ - {x^3}}}} dx$$

For x $$ \in $$ (0, 1)

$$ \Rightarrow $$  x > x² or $$-$$ x < $$-$$ x²

and x² > x³ or $$-$$ x² < $$-$$ x³

$$ \therefore $$ $${e^{ - {x^2}}}$$ < $${e^{ - {x^3}}}$$ and $${e^{ - x}}$$ < $${e^{ - {x^2}}}$$

$$ \Rightarrow $$ $${e^{ - x}} < {e^{ - {x^2}}} < {e^{ - {x^3}}}$$

$$ \Rightarrow $$ $${e^{ - {x^3}}} > {e^{ - {x^2}}} > {e^{ - x}}$$

$$ \Rightarrow $$ $${I_3} > {I_2} > {I_1}$$