`int_(0)^(9) x(a^(2)-x^(2))^(7/2)dx`

Answer

To solve the integral \( I = \int_{0}^{9} x (a^2 - x^2)^{7/2} \, dx \), we will use the substitution method. Let's go through the steps:

### Step 1: Substitution
Let \( T = a^2 - x^2 \). Then, we differentiate \( T \) with respect to \( x \):
\[ dT = -2x \, dx \quad \Rightarrow \quad dx = -\frac{dT}{2x} \]

### Step 2: Change the limits of integration
When \( x = 0 \):
\[ T = a^2 - 0^2 = a^2 \]
When \( x = 9 \):
\[ T = a^2 - 9^2 = a^2 - 81 \]

### Step 3: Substitute in the integral
Now we can rewrite the integral \( I \):
\[ I = \int_{a^2}^{a^2 - 81} x (T)^{7/2} \left(-\frac{dT}{2x}\right) \]
The \( x \) terms cancel out:
\[ I = -\frac{1}{2} \int_{a^2}^{a^2 - 81} (T)^{7/2} \, dT \]

### Step 4: Change the limits of integration
We can switch the limits of integration, which changes the sign:
\[ I = \frac{1}{2} \int_{a^2 - 81}^{a^2} (T)^{7/2} \, dT \]

### Step 5: Integrate
Now we integrate \( (T)^{7/2} \):
\[ \int (T)^{7/2} \, dT = \frac{T^{9/2}}{9/2} = \frac{2}{9} T^{9/2} \]
Thus,
\[ I = \frac{1}{2} \cdot \frac{2}{9} \left[ T^{9/2} \right]_{a^2 - 81}^{a^2} \]
This simplifies to:
\[ I = \frac{1}{9} \left[ (a^2)^{9/2} - (a^2 - 81)^{9/2} \right] \]

### Step 6: Final result
Now, substituting back:
\[ I = \frac{1}{9} \left[ a^9 - (a^2 - 81)^{9/2} \right] \]

### Conclusion
Thus, the final value of the integral is:
\[ I = \frac{1}{9} \left[ a^9 - (a^2 - 81)^{9/2} \right] \]