Evaluate :
`int_(4)^(9)(sqrt(x))/(30-x^(3//2))^2dx`

Answer

To evaluate the integral

\[ \int_{4}^{9} \frac{\sqrt{x}}{(30 - x^{3/2})^2} \, dx, \]

we will use a substitution method. Let's go through the steps systematically.

### Step 1: Substitution
Let

\[ t = 30 - x^{3/2}. \]

Now, we need to find \(dx\) in terms of \(dt\). First, we differentiate \(t\) with respect to \(x\):

\[ \frac{dt}{dx} = -\frac{3}{2} x^{1/2}. \]

Thus, we can express \(dx\) as:

\[ dx = -\frac{2}{3} \frac{dt}{\sqrt{x}}. \]

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

- When \(x = 4\):
\[ t = 30 - 4^{3/2} = 30 - 8 = 22. \]

- When \(x = 9\):
\[ t = 30 - 9^{3/2} = 30 - 27 = 3. \]

So, the new limits for \(t\) will be from \(22\) to \(3\).

### Step 3: Substitute in the Integral
Now we substitute \(t\) and \(dx\) into the integral:

\[ \int_{4}^{9} \frac{\sqrt{x}}{(30 - x^{3/2})^2} \, dx = \int_{22}^{3} \frac{\sqrt{x}}{t^2} \left(-\frac{2}{3} \frac{dt}{\sqrt{x}}\right). \]

The \(\sqrt{x}\) terms cancel out:

\[ = -\frac{2}{3} \int_{22}^{3} \frac{1}{t^2} \, dt. \]

### Step 4: Change the Limits
We can change the limits of integration, which introduces a negative sign:

\[ = \frac{2}{3} \int_{3}^{22} \frac{1}{t^2} \, dt. \]

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

\[ \int \frac{1}{t^2} \, dt = -\frac{1}{t}. \]

Thus,

\[ \frac{2}{3} \left[-\frac{1}{t}\right]_{3}^{22} = \frac{2}{3} \left(-\frac{1}{22} + \frac{1}{3}\right). \]

### Step 6: Simplify
Calculating the expression:

\[ -\frac{1}{22} + \frac{1}{3} = \frac{-3 + 22}{66} = \frac{19}{66}. \]

So we have:

\[ \frac{2}{3} \cdot \frac{19}{66} = \frac{38}{198} = \frac{19}{99}. \]

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

\[ \frac{19}{99}. \]

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