`int((9x^(2)-4x+5))/((3x^(3)-2x^(2)+5x+1))dx`

Answer

To solve the integral

\[ \int \frac{9x^2 - 4x + 5}{3x^3 - 2x^2 + 5x + 1} \, dx, \]

we will use the method of integration by substitution. Here are the steps:

### Step 1: Identify the Denominator
Let

\[ T = 3x^3 - 2x^2 + 5x + 1. \]

### Step 2: Differentiate T
Now, we need to find the derivative of T with respect to x:

\[ \frac{dT}{dx} = \frac{d}{dx}(3x^3 - 2x^2 + 5x + 1) = 9x^2 - 4x + 5. \]

### Step 3: Rewrite the Integral
We can rewrite the integral using the substitution \( T \):

\[ \int \frac{9x^2 - 4x + 5}{3x^3 - 2x^2 + 5x + 1} \, dx = \int \frac{dT}{T}. \]

### Step 4: Integrate
The integral of \(\frac{dT}{T}\) is:

\[ \int \frac{dT}{T} = \ln |T| + C, \]

where C is the constant of integration.

### Step 5: Substitute Back T
Now, substitute back the expression for T:

\[ \int \frac{9x^2 - 4x + 5}{3x^3 - 2x^2 + 5x + 1} \, dx = \ln |3x^3 - 2x^2 + 5x + 1| + C. \]

### Final Answer
Thus, the final result is:

\[ \ln |3x^3 - 2x^2 + 5x + 1| + C. \]

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