6. Evaluate the indefinite integral. (Use \( C \) for the constant of integration.) \[ \int \frac{x^{9}}{1+x^{20}} d x \]

First, let's consider the substitution \( u = 1 + x^{20} \). This implies that \( du = 20x^{19} \, dx \), which allows us to express \( dx \) in terms of \( du \): \[ dx = \frac{du}{20x^{19}}. \] Now, we need to express \( x^9 \) in terms of \( u \). Since \( u = 1 + x^{20} \), we can solve for \( x^{20} \): \[ x^{20} = u - 1 \quad \Rightarrow \quad x^9 = (u - 1)^{\frac{9}{20}}. \] This replacement can lead to complicated computations, so instead, let's first focus on the structure of the integral. Notice that we can use the properties of differentiation and reintegration to help simplify this integral. The structure \( \frac{x^{9}}{1+x^{20}} \) suggests using a polynomial long division or recognizing a potential arctangent structure. Instead, we can use the following **trick**: Let's make the observation that the integral \( \int \frac{x^{9}}{1+x^{20}} \, dx \) behaves nicely by considering a series expansion for \( \frac{1}{1+x^{20}} \) around \( x = 0 \). This brings about a series of simpler integrals. However, this is an integral that does not yield a simple expression in terms of elementary functions, so we express the solution as: \[ \int \frac{x^{9}}{1+x^{20}} \, dx = \frac{1}{20} \ln(1+x^{20}) + C. \] And there we have our indefinite integral with the constant of integration \( C \).