Certainly! Let's evaluate the integral [tex]\(\int \frac{x + 8^x}{x - 5} \, dx\)[/tex].
This is a rational function where both the numerator and the denominator involve the variable [tex]\(x\)[/tex]. However, the expression [tex]\(8^x\)[/tex] in the numerator makes it a bit more complex since it's an exponential term.
Given the integral:
[tex]\[
\int \frac{x + 8^x}{x - 5} \, dx
\][/tex]
We need to find the antiderivative of this function. In general, finding an antiderivative for such a function may not yield a simple elementary function. The antiderivative is expressed in terms of an integral that cannot be simplified further easily.
The result of this integral, as it turns out, is left in its integral form for further simplification or numerical computation, and thus we have:
[tex]\[
\int \frac{x + 8^x}{x - 5} \, dx = \int \frac{x + 8^x}{x - 5} \, dx
\][/tex]
This is the most simplified form of the expression for the integral and it represents the antiderivative.
