General antiderivative of an exponential plus power function

Split the integral using linearity: $\int (5^x + x^5)\,dx = \int 5^x\,dx + \int x^5\,dx$. Each term can be antidifferentiated separately using known rules.

For the exponential term, recall $\int a^x\,dx = \dfrac{a^x}{\ln a} + C$, so $\int 5^x\,dx = \dfrac{5^x}{\ln 5}$. For the power term, reverse the power rule: $\int x^5\,dx = \dfrac{x^6}{6}$. Combining these and adding a single arbitrary constant gives $F(x) = \dfrac{5^x}{\ln 5} + \dfrac{x^6}{6} + C$.