Question: Evaluate the following integral: ∫(x^3 + 2x^2 + 3x + 4) / (x^4 + 4x^3 + 6x^2 + 4x + 1) dx solve it

1. Break Down the Problem

We need to evaluate the integral

$I = \int \frac{x^3 + 2x^2 + 3x + 4}{x^4 + 4x^3 + 6x^2 + 4x + 1} \, dx$

2. Relevant Concepts

To solve this integral, we might start by examining the denominator. We see that the polynomial $x^4 + 4x^3 + 6x^2 + 4x + 1$ could be related to a square of a quadratic polynomial.

3. Analysis and Detail

Let's first rewrite the denominator:

$x^4 + 4x^3 + 6x^2 + 4x + 1 = (x^2 + 2x + 1)^2 = (x + 1)^4$

Thus, the integral simplifies to:

$I = \int \frac{x^3 + 2x^2 + 3x + 4}{(x + 1)^4} \, dx$

Next, we can perform polynomial long division to separate the integral into simpler parts if needed. Start by rewriting the numerator in terms of the denominator:

1. Divide $x^3 + 2x^2 + 3x + 4$ by $x + 1$:

$x^3 + 2x^2 + 3x + 4 = (x^2 + x + 2)(x + 1) + 2$

Thus, we can rewrite the integral:

$I = \int \left( \frac{x^2 + x + 2}{(x+1)^3} + \frac{2}{(x+1)^4} \right) \, dx$

This gives us two integrals to solve:

$I = \int \frac{x^2 + x + 2}{(x+1)^3} \, dx + \int \frac{2}{(x+1)^4} \, dx$

4. Verify and Summarize

Now we can evaluate each integral separately:

1. For $\int \frac{2}{(x + 1)^4} \, dx$:

$\int \frac{2}{(x + 1)^4} \, dx = -\frac{2}{3(x + 1)^3} + C_1$

2. For $\int \frac{x^2 + x + 2}{(x + 1)^3} \, dx$, we can apply the substitution $u = x + 1$ which simplifies the calculation.

$= \int \frac{(u-1)^2 + (u-1) + 2}{u^3} \, du = \int \frac{u^2 - 2u + 1 + u - 1 + 2}{u^3} \, du = \int \frac{u^2 - u + 2}{u^3} \, du$

Break it down further:

$= \int \left( \frac{1}{u} - \frac{1}{u^2} + \frac{2}{u^3} \right) \, du$

Evaluating this gives:

$= \ln |u| + \frac{1}{u} - \frac{2}{2u^2} + C_2 = \ln |x + 1| - \frac{1}{(x + 1)^2} + C_2$

Combining these results:

$I = \left[ \ln |x + 1| - \frac{1}{(x + 1)^2} - \frac{2}{3(x + 1)^3} + C \right]$

Final Answer

Thus, the evaluated integral is:

$I = \ln |x + 1| - \frac{1}{(x + 1)^2} - \frac{2}{3(x + 1)^3} + C$ where $C$ is the constant of integration.