Question Evaluate the integral $\int_{e^{3 x}}^{e^{4 x}} \frac{1}{x \ln x+x} dx$.

Studdy Solution
Substitute back for $v$ and $w$ to get $F(x)$ in terms of $x$.
$$F(x) = \frac{1}{4} \ln|4x + 1| - \frac{1}{3} \ln|3x + 1| + C$$
This function $F(x)$ represents the antiderivative of the original integral. However, we must acknowledge that the original problem posed an improper integral with variable limits of integration, which cannot be evaluated using standard methods. The steps provided here lead to an antiderivative of a related function, but they do not solve the original problem as stated. The original integral requires a more sophisticated approach or clarification of the problem statement.