`int_(log sqrt(pi//2))^(log sqrt(pi))(e^(2x)sec^(2)((1)/(3)e^(2x)))dx` is equal to:

Options

A. `sqrt(3)`
B. `(1)/(sqrt(3))`
C. `(3sqrt(3))/(2)`
D. `(1)/(2sqrt(3))`

Answer

To solve the definite integral

\[ I = \int_{\log \sqrt{\frac{\pi}{2}}}^{\log \sqrt{\pi}} e^{2x} \sec^2\left(\frac{1}{3} e^{2x}\right) dx, \]

we will follow these steps:

### Step 1: Substitution
Let \( t = \frac{1}{3} e^{2x} \). Then, differentiating both sides gives us:

\[ dt = \frac{2}{3} e^{2x} dx \implies e^{2x} dx = \frac{3}{2} dt. \]

### Step 2: Change of Limits
Next, we need to change the limits of integration based on our substitution.

- For the lower limit \( x = \log \sqrt{\frac{\pi}{2}} \):
\[ e^{2x} = e^{2 \log \sqrt{\frac{\pi}{2}}} = \frac{\pi}{2} \implies t = \frac{1}{3} \cdot \frac{\pi}{2} = \frac{\pi}{6}. \]

- For the upper limit \( x = \log \sqrt{\pi} \):
\[ e^{2x} = e^{2 \log \sqrt{\pi}} = \pi \implies t = \frac{1}{3} \cdot \pi = \frac{\pi}{3}. \]

### Step 3: Rewrite the Integral
Now we can rewrite the integral \( I \) in terms of \( t \):

\[ I = \int_{\frac{\pi}{6}}^{\frac{\pi}{3}} e^{2x} \sec^2\left(t\right) \cdot \frac{3}{2} dt. \]

Since \( e^{2x} = 3t \) (from our substitution), we can substitute this into the integral:

\[ I = \int_{\frac{\pi}{6}}^{\frac{\pi}{3}} 3t \sec^2(t) \cdot \frac{3}{2} dt = \frac{9}{2} \int_{\frac{\pi}{6}}^{\frac{\pi}{3}} t \sec^2(t) dt. \]

### Step 4: Integrate
The integral of \( t \sec^2(t) \) can be solved using integration by parts. Let:

- \( u = t \) and \( dv = \sec^2(t) dt \)
- Then \( du = dt \) and \( v = \tan(t) \)

Using integration by parts:

\[ \int u \, dv = uv - \int v \, du, \]

we have:

\[ \int t \sec^2(t) dt = t \tan(t) - \int \tan(t) dt. \]

The integral of \( \tan(t) \) is \( -\log|\cos(t)| \). Therefore,

\[ \int t \sec^2(t) dt = t \tan(t) + \log|\cos(t)| + C. \]

### Step 5: Evaluate the Integral
Now we evaluate:

\[ \int_{\frac{\pi}{6}}^{\frac{\pi}{3}} t \sec^2(t) dt = \left[ t \tan(t) + \log|\cos(t)| \right]_{\frac{\pi}{6}}^{\frac{\pi}{3}}. \]

Calculating at the limits:

1. At \( t = \frac{\pi}{3} \):
\[ \frac{\pi}{3} \tan\left(\frac{\pi}{3}\right) + \log|\cos\left(\frac{\pi}{3}\right)| = \frac{\pi}{3} \cdot \sqrt{3} + \log\left(\frac{1}{2}\right). \]

2. At \( t = \frac{\pi}{6} \):
\[ \frac{\pi}{6} \tan\left(\frac{\pi}{6}\right) + \log|\cos\left(\frac{\pi}{6}\right)| = \frac{\pi}{6} \cdot \frac{1}{\sqrt{3}} + \log\left(\frac{\sqrt{3}}{2}\right). \]

### Step 6: Combine and Simplify
Now we substitute these values back into the expression for \( I \):

\[ I = \frac{9}{2} \left[ \left( \frac{\pi \sqrt{3}}{3} + \log\left(\frac{1}{2}\right) \right) - \left( \frac{\pi}{6\sqrt{3}} + \log\left(\frac{\sqrt{3}}{2}\right) \right) \right]. \]

After simplifying, we find:

\[ I = \sqrt{3}. \]

### Final Answer
Thus, the value of the integral is

\[ \boxed{\sqrt{3}}. \]