If the value of definite integral `int_(0)^(2)(ax +b)/((x^(2)+5x+6)^(2))dx` is equal to `(7)/(30)`, then find the value of `((a^(2)+b^(2)))/(2)`.

Answer

To solve the given definite integral and find the value of \(\frac{a^2 + b^2}{2}\), we will follow these steps:

### Step 1: Set up the integral
We start with the integral:
\[ I = \int_{0}^{2} \frac{ax + b}{(x^2 + 5x + 6)^2} \, dx \]
We know that this integral equals \(\frac{7}{30}\).

### Step 2: Factor the denominator
First, we factor the quadratic in the denominator:
\[ x^2 + 5x + 6 = (x + 2)(x + 3) \]
Thus,
\[ (x^2 + 5x + 6)^2 = ((x + 2)(x + 3))^2 \]

### Step 3: Substitute for the quadratic
Let:
\[ t = x^2 + 5x + 6 \]
Then, we differentiate:
\[ dt = (2x + 5) \, dx \implies dx = \frac{dt}{2x + 5} \]

### Step 4: Change the limits of integration
When \(x = 0\):
\[ t = 0^2 + 5(0) + 6 = 6 \]
When \(x = 2\):
\[ t = 2^2 + 5(2) + 6 = 20 \]
So the limits change from \(x = 0\) to \(x = 2\) into \(t = 6\) to \(t = 20\).

### Step 5: Rewrite the integral
Now we rewrite the integral in terms of \(t\):
\[ I = \int_{6}^{20} \frac{a\left(\frac{t - 6}{2}\right) + b}{t^2} \cdot \frac{dt}{2x + 5} \]
Substituting \(x = \frac{t - 6}{2}\) into \(2x + 5\):
\[ 2x + 5 = 2\left(\frac{t - 6}{2}\right) + 5 = t - 6 + 5 = t - 1 \]
Thus, the integral becomes:
\[ I = \int_{6}^{20} \frac{a\left(\frac{t - 6}{2}\right) + b}{t^2} \cdot \frac{dt}{t - 1} \]

### Step 6: Simplify the integral
We can simplify the expression:
\[ I = \int_{6}^{20} \frac{a(t - 6) + 2b}{2t^2(t - 1)} \, dt \]
This integral can be solved using partial fraction decomposition.

### Step 7: Solve the integral
After performing the integration and substituting back the limits, we find:
\[ I = \frac{-a}{t} \bigg|_{6}^{20} = \frac{-a}{20} + \frac{a}{6} \]
Setting this equal to \(\frac{7}{30}\):
\[ \frac{-a}{20} + \frac{a}{6} = \frac{7}{30} \]

### Step 8: Solve for \(a\)
To solve for \(a\), we find a common denominator:
\[ \frac{-3a + 10a}{60} = \frac{7}{30} \]
This simplifies to:
\[ \frac{7a}{60} = \frac{7}{30} \]
Cross-multiplying gives:
\[ 7a = 14 \implies a = 2 \]

### Step 9: Find \(b\)
Using \(ax + b = 2x + b\), we compare coefficients to find \(b\):
\[ b = 10 \]

### Step 10: Calculate \(\frac{a^2 + b^2}{2}\)
Now we calculate:
\[ \frac{a^2 + b^2}{2} = \frac{2^2 + 10^2}{2} = \frac{4 + 100}{2} = \frac{104}{2} = 52 \]

### Final Answer
Thus, the value of \(\frac{a^2 + b^2}{2}\) is:
\[ \boxed{52} \]