The image shows a set of integrals involving rational functions, titled "Fungsi Rasional: Pembagian Polinom" (Rational Functions: Polynomial Division). These integrals are likely part of a section on the integration of rational functions, which can often be solved through polynomial division or partial fraction decomposition.

Let's start with the solutions for the integrals.

1. $\int \frac{x^2 + 2x + 1}{x - 1} \, dx$

This is a rational function where the degree of the numerator is higher than the degree of the denominator. We can use polynomial division to simplify it.

1. Perform polynomial division: $\frac{x^2 + 2x + 1}{x - 1} = x + 3 + \frac{4}{x - 1}$
2. Now integrate term by term: $\int \left( x + 3 + \frac{4}{x - 1} \right) dx = \frac{x^2}{2} + 3x + 4 \ln |x - 1| + C$

2. $\int \frac{x^3 + 3x^2 + 2x}{x^2 - 1} \, dx$

This can be simplified by first performing polynomial division:

1. Factor the denominator: $x^2 - 1 = (x - 1)(x + 1)$
2. Perform the division: $\frac{x^3 + 3x^2 + 2x}{x^2 - 1} = x + 4 + \frac{4x + 4}{x^2 - 1}$
3. Decompose the remaining fraction: $\frac{4x + 4}{x^2 - 1} = \frac{2}{x - 1} + \frac{2}{x + 1}$
4. Integrate each term: $\int \left( x + 4 + \frac{2}{x - 1} + \frac{2}{x + 1} \right) dx = \frac{x^2}{2} + 4x + 2 \ln |x - 1| + 2 \ln |x + 1| + C$

3. $\int \frac{x^4 + 4x^3 + 3x^2}{x^2 + 1} \, dx$

We can again start by performing polynomial division.

1. Perform polynomial division: $\frac{x^4 + 4x^3 + 3x^2}{x^2 + 1} = x^2 + 4x + 2 - \frac{2}{x^2 + 1}$
2. Now integrate term by term: $\int \left( x^2 + 4x + 2 - \frac{2}{x^2 + 1} \right) dx = \frac{x^3}{3} + 2x^2 + 2x - 2 \tan^{-1}(x) + C$

Would you like to proceed with solving the remaining integrals, or do you have any questions on these solutions?

Follow-up questions:

1. Can you explain the method of polynomial division in integrals?
2. How does partial fraction decomposition help with integrals like these?
3. Can you solve integrals with irreducible quadratics in the denominator?
4. What is the difference between indefinite and definite integrals?
5. How can logarithmic functions appear in integrals involving rational functions?

Tip:

When solving rational function integrals, always check the degrees of the numerator and denominator to decide if polynomial division or partial fraction decomposition is necessary.