Verify that`int(2x+3)/(x^(2)+3x)dx = log|x^(2)+3x|+c`

Explanation

To verify the integral \(\int \frac{2x+3}{x^2+3x} \, dx = \log|x^2+3x| + C\), we will follow these steps:

### Step 1: Identify the integral
We start with the left-hand side (LHS):
\[ I = \int \frac{2x + 3}{x^2 + 3x} \, dx \]

### Step 2: Simplify the denominator
We notice that the denominator can be factored:
\[ x^2 + 3x = x(x + 3) \]

### Step 3: Use substitution
We will use the substitution method. Let:
\[ t = x^2 + 3x \]
Now, we need to find \(dt\):
\[ \frac{dt}{dx} = \frac{d}{dx}(x^2 + 3x) = 2x + 3 \]
Thus, we have:
\[ dt = (2x + 3) \, dx \]

### Step 4: Rewrite the integral
Now, substituting \(t\) and \(dt\) into the integral:
\[ I = \int \frac{dt}{t} \]

### Step 5: Integrate
The integral of \(\frac{1}{t}\) is:
\[ I = \log |t| + C \]

### Step 6: Substitute back for \(t\)
Now, we substitute back \(t = x^2 + 3x\):
\[ I = \log |x^2 + 3x| + C \]

### Step 7: Conclusion
Thus, we have verified that:
\[ \int \frac{2x + 3}{x^2 + 3x} \, dx = \log |x^2 + 3x| + C \]