Evaluate `int_(1)^(5)sqrt(x-2)sqrt(x-1)dx`.

Explanation

To evaluate the integral \( I = \int_{1}^{5} \sqrt{x-2} \sqrt{x-1} \, dx \), we will follow these steps:

### Step 1: Simplify the integrand
We can rewrite the integrand as:
\[ I = \int_{1}^{5} \sqrt{(x-2)(x-1)} \, dx \]

### Step 2: Identify the critical points
The expression under the square root, \((x-2)(x-1)\), becomes negative when \(x < 2\). Therefore, we need to split the integral at the point where the expression becomes zero, which is at \(x = 2\).

### Step 3: Split the integral
We can split the integral into two parts:
\[ I = \int_{1}^{2} \sqrt{(x-2)(x-1)} \, dx + \int_{2}^{5} \sqrt{(x-2)(x-1)} \, dx \]

### Step 4: Handle the negative values
For \(x\) in the interval \([1, 2]\), \((x-2)(x-1)\) is negative. Thus, we take the negative of the square root:
\[ I = \int_{1}^{2} -\sqrt{(2-x)(x-1)} \, dx + \int_{2}^{5} \sqrt{(x-2)(x-1)} \, dx \]

### Step 5: Evaluate the first integral
Let’s evaluate the first integral:
\[ \int_{1}^{2} -\sqrt{(2-x)(x-1)} \, dx \]
We can use the substitution \(u = 2 - x\), which gives \(du = -dx\) and changes the limits from \(x=1\) to \(u=1\) and from \(x=2\) to \(u=0\):
\[ \int_{1}^{2} -\sqrt{(2-x)(x-1)} \, dx = \int_{0}^{1} \sqrt{u(1-u)} \, du \]

### Step 6: Evaluate the second integral
For the second integral:
\[ \int_{2}^{5} \sqrt{(x-2)(x-1)} \, dx \]
We can use the substitution \(v = x - 2\), which gives \(dv = dx\) and changes the limits from \(x=2\) to \(v=0\) and from \(x=5\) to \(v=3\):
\[ \int_{2}^{5} \sqrt{(x-2)(x-1)} \, dx = \int_{0}^{3} \sqrt{v(v+1)} \, dv \]

### Step 7: Combine the integrals
Now we have:
\[ I = \int_{0}^{1} \sqrt{u(1-u)} \, du + \int_{0}^{3} \sqrt{v(v+1)} \, dv \]

### Step 8: Evaluate the integrals
The integral \(\int_{0}^{1} \sqrt{u(1-u)} \, du\) can be evaluated using the Beta function or trigonometric substitution, and it equals \(\frac{\pi}{8}\).

The integral \(\int_{0}^{3} \sqrt{v(v+1)} \, dv\) can be evaluated using integration techniques, and it results in a value that can be computed numerically or using a definite integral table.

### Step 9: Final calculation
After evaluating both integrals, we can combine the results to find the total value of \(I\).