Consider the integrals
`I_(1)=overset(1)underset(0)inte^(-x)cos^(2)xdx,I_(2)=overset(1)underset(0)int e^(-x^(2))cos^(2)x dx,I_(3)=overset(1)underset(0)int e^(-x^(2//2))cos^(2)xdx`
and `I_(4)=overset(1)underset(0)int e^(-x^(2//2))dx`. The greatest of these integrals, is

Options

A. `I_(1)lt I_(2)lt I_(3)`
B. `I_(3)lt I_(2)lt I_(1)`
C. `I_(2)lt I_(1)lt I_(3)`
D. `I_(2)lt I_(3)lt I_(1)`

Answer

To solve the problem, we need to compare the integrals \( I_1, I_2, I_3, \) and \( I_4 \) defined as follows:

\[ I_1 = \int_0^1 e^{-x} \cos^2 x \, dx \]
\[ I_2 = \int_0^1 e^{-x^2} \cos^2 x \, dx \]
\[ I_3 = \int_0^1 e^{-\frac{x^2}{2}} \cos^2 x \, dx \]
\[ I_4 = \int_0^1 e^{-\frac{x^2}{2}} \, dx \]

### Step 1: Establishing Inequalities
We will analyze the behavior of the exponential functions in each integral. Notice that for \( x \) in the interval \( [0, 1] \):

- \( e^{-x} \) is decreasing.
- \( e^{-x^2} \) is decreasing and greater than \( e^{-\frac{x^2}{2}} \) because \( -x^2 < -\frac{x^2}{2} \) for \( x > 0 \).
- \( e^{-\frac{x^2}{2}} \) is also decreasing.

From the above observations, we can establish the following inequalities:

\[ e^{-x} < e^{-x^2} < e^{-\frac{x^2}{2}} \quad \text{for } x \in [0, 1] \]

### Step 2: Multiplying by \( \cos^2 x \)
Since \( \cos^2 x \) is non-negative for \( x \in [0, 1] \), we can multiply the inequalities by \( \cos^2 x \):

\[ e^{-x} \cos^2 x < e^{-x^2} \cos^2 x < e^{-\frac{x^2}{2}} \cos^2 x \quad \text{for } x \in [0, 1] \]

### Step 3: Integrating the Inequalities
Now we integrate the inequalities over the interval from 0 to 1:

\[ \int_0^1 e^{-x} \cos^2 x \, dx < \int_0^1 e^{-x^2} \cos^2 x \, dx < \int_0^1 e^{-\frac{x^2}{2}} \cos^2 x \, dx \]

This gives us:

\[ I_1 < I_2 < I_3 \]

### Step 4: Comparing \( I_3 \) and \( I_4 \)
Next, we compare \( I_3 \) and \( I_4 \). Since \( \cos^2 x \leq 1 \) for all \( x \), we have:

\[ I_3 = \int_0^1 e^{-\frac{x^2}{2}} \cos^2 x \, dx \leq \int_0^1 e^{-\frac{x^2}{2}} \, dx = I_4 \]

Thus, we can conclude:

\[ I_1 < I_2 < I_3 < I_4 \]

### Conclusion
The greatest of these integrals is:

\[ \boxed{I_4} \]