Evaluate the following integrals :
(a) `int (dx)/( x^(2) - 6x + 13),`

Answer

To evaluate the integral

\[ \int \frac{dx}{x^2 - 6x + 13}, \]

we can follow these steps:

### Step 1: Complete the square in the denominator

The expression in the denominator is \(x^2 - 6x + 13\). We can complete the square for the quadratic expression:

\[ x^2 - 6x + 13 = (x^2 - 6x + 9) + 4 = (x - 3)^2 + 2^2. \]

### Step 2: Rewrite the integral

Now we can rewrite the integral using the completed square:

\[ \int \frac{dx}{(x - 3)^2 + 2^2}. \]

### Step 3: Use the standard integral formula

We can use the standard integral formula:

\[ \int \frac{dx}{x^2 + a^2} = \frac{1}{a} \tan^{-1} \left( \frac{x}{a} \right) + C. \]

In our case, we have \(a = 2\) and \(x\) is replaced by \(x - 3\). Thus, we can apply the formula:

\[ \int \frac{dx}{(x - 3)^2 + 2^2} = \frac{1}{2} \tan^{-1} \left( \frac{x - 3}{2} \right) + C. \]

### Step 4: Write the final answer

So, the final answer for the integral is:

\[ \int \frac{dx}{x^2 - 6x + 13} = \frac{1}{2} \tan^{-1} \left( \frac{x - 3}{2} \right) + C. \]

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