`Evaluate : int(1)/(5+4x+x^(2))dx`

Answer

To evaluate the integral \( I = \int \frac{1}{5 + 4x + x^2} \, dx \), we can follow these steps:

### Step 1: Rewrite the Denominator
First, we need to rewrite the quadratic expression in the denominator. We can complete the square for the expression \( x^2 + 4x + 5 \).

\[ x^2 + 4x + 5 = (x^2 + 4x + 4) + 1 = (x + 2)^2 + 1 \]

### Step 2: Substitute the Completed Square into the Integral
Now we can substitute this back into the integral:

\[ I = \int \frac{1}{(x + 2)^2 + 1} \, dx \]

### Step 3: Use the Standard Integral Formula
The integral we have now resembles the standard integral form:

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

In our case, \( a^2 = 1 \) so \( a = 1 \) and \( x \) is replaced by \( x + 2 \).

### Step 4: Apply the Formula
Using the standard formula, we get:

\[ I = \int \frac{1}{(x + 2)^2 + 1} \, dx = \tan^{-1}(x + 2) + C \]

### Final Answer
Thus, the evaluated integral is:

\[ I = \tan^{-1}(x + 2) + C \]

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