2–74. Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals.
10. ∫ (x³ + 3x² + 1)/(x³ + 1) dx

Verified step by step guidance

1. Step 1: Simplify the integrand by performing polynomial long division. Divide \(x^3 + 3x^2 + 1\) by \(x^3 + 1\) to rewrite the integrand as a sum of simpler terms.
2. Step 2: After performing the division, express the integrand as \(Q(x) + \frac{R(x)}{x^3 + 1}\), where \(Q(x)\) is the quotient and \(R(x)\) is the remainder. This will simplify the integral into separate terms.
3. Step 3: Factorize \(x^3 + 1\) in the denominator of the fractional term. Recall that \(x^3 + 1 = (x + 1)(x^2 - x + 1)\). This will help in breaking down the fractional term further.
4. Step 4: Split the integral into separate parts: one for the polynomial quotient \(Q(x)\) and another for the fractional term \(\frac{R(x)}{(x + 1)(x^2 - x + 1)}\). Use partial fraction decomposition to handle the fractional term.
5. Step 5: Integrate each term separately. For the polynomial term \(Q(x)\), use basic power rule integration. For the fractional term, integrate using the results of the partial fraction decomposition, which may involve logarithmic and arctangent functions.

Key Concepts

Integration Techniques

Integration techniques are methods used to find the integral of a function. Common techniques include substitution, integration by parts, and partial fraction decomposition. Understanding these methods is crucial for evaluating complex integrals, as they allow for simplification and easier computation.

Polynomial Long Division

Polynomial long division is a method used to divide one polynomial by another. In the context of integrals, it is often applied when the degree of the numerator is greater than or equal to the degree of the denominator. This technique simplifies the integrand into a more manageable form, making it easier to integrate.

Rational Functions

A rational function is a ratio of two polynomials. When integrating rational functions, it is important to analyze their behavior and potential simplifications. Techniques such as partial fraction decomposition can be employed to break down complex rational functions into simpler components that are easier to integrate.