General antiderivative of a polynomial-exponential-reciprocal function

Integrate term by term using linearity. For $10x^4$, use the power rule: $\int 10x^4\,dx = 10\cdot \frac{x^5}{5} = 2x^5$. For $-e^x$, recall $\int e^x\,dx = e^x$, so this term contributes $-e^x$. For $\frac{1}{x}$, recall $\int \frac{1}{x}\,dx = \ln|x|$.

Combining these results gives $2x^5 - e^x + \ln|x|$. Since the derivative of any constant is zero, we must add an arbitrary constant $C$ to represent the entire family of antiderivatives. The final general antiderivative is $F(x) = 2x^5 - e^x + \ln|x| + C$.