Evaluate the integrals in Exercises 69–134. The integrals are listed in random order so you need to decide which integration technique to use.
121. ∫ (1 + x²) / (1 + x³) dx

Verified step by step guidance

1. First, examine the integrand \( \frac{1 + x^{2}}{1 + x^{3}} \) to see if it can be simplified or if a substitution is appropriate.
2. Notice that the denominator \(1 + x^{3}\) can be factored using the sum of cubes formula: \(a^{3} + b^{3} = (a + b)(a^{2} - ab + b^{2})\). So, \(1 + x^{3} = (x + 1)(x^{2} - x + 1)\).
3. Try to express the numerator \(1 + x^{2}\) in terms of the factors of the denominator or consider polynomial division if the degree of numerator is less than denominator.
4. Alternatively, consider the substitution \(u = 1 + x^{3}\), then compute \(du = 3x^{2} dx\). This suggests rewriting the integral in terms of \(u\) if possible.
5. If substitution is not straightforward, try to decompose the integrand into partial fractions based on the factorization of the denominator and then integrate each term separately.

Key Concepts

Integration Techniques

Integration techniques are methods used to find antiderivatives of functions. Common techniques include substitution, partial fractions, integration by parts, and trigonometric substitution. Choosing the right technique depends on the form of the integrand and often requires algebraic manipulation.

Algebraic Manipulation of Rational Functions

Rational functions are ratios of polynomials. Simplifying or decomposing these functions, such as by polynomial division or partial fraction decomposition, helps in applying integration techniques effectively. Recognizing factorization patterns is key to simplifying the integral.

Substitution Method

The substitution method involves changing variables to simplify an integral. By identifying a part of the integrand whose derivative also appears, you can substitute it with a new variable, transforming the integral into a simpler form that is easier to evaluate.