To solve the problem, we need to analyze the integrals \( I_1 \), \( I_2 \), and \( I_3 \) defined as follows:
\[
I_1 = \int_{0}^{\frac{\pi}{4}} x^{2008} (\tan x)^{2008} \, dx
\]
\[
I_2 = \int_{0}^{\frac{\pi}{4}} x^{2009} (\tan x)^{2009} \, dx
\]
\[
I_3 = \int_{0}^{\frac{\pi}{4}} x^{2010} (\tan x)^{2010} \, dx
\]
### Step 1: Analyze the behavior of \( \tan x \)
For \( x \) in the interval \( [0, \frac{\pi}{4}] \):
- \( \tan x \) is a continuous and increasing function.
- At \( x = 0 \), \( \tan 0 = 0 \).
- At \( x = \frac{\pi}{4} \), \( \tan \frac{\pi}{4} = 1 \).
Thus, \( 0 \leq \tan x < 1 \) for \( x \in [0, \frac{\pi}{4}) \).
### Step 2: Compare the powers of \( x \) and \( \tan x \)
Since \( \tan x < 1 \) in the interval \( [0, \frac{\pi}{4}) \), we can analyze the behavior of the integrands:
- For \( x^{2008} (\tan x)^{2008} \), since both \( x \) and \( \tan x \) are raised to positive powers, we have:
\[
x^{2008} (\tan x)^{2008} < x^{2009} (\tan x)^{2009} < x^{2010} (\tan x)^{2010}
\]
because \( x < x^{2009} < x^{2010} \) and \( \tan x < 1 \).
### Step 3: Integrate the inequalities
Using the property of integrals, we can integrate the inequalities over the interval \( [0, \frac{\pi}{4}] \):
\[
\int_{0}^{\frac{\pi}{4}} x^{2008} (\tan x)^{2008} \, dx < \int_{0}^{\frac{\pi}{4}} x^{2009} (\tan x)^{2009} \, dx < \int_{0}^{\frac{\pi}{4}} x^{2010} (\tan x)^{2010} \, dx
\]
This gives us the relationships:
\[
I_1 < I_2 < I_3
\]
### Conclusion
Thus, the correct inequality is:
\[
I_3 < I_2 < I_1
\]
