Calculate $\int 5^{x+1}e^{2x-1}\,dx$

$$\int 5^{x+1}e^{2x-1}dx=\int e^{(x+1)\ln 5}e^{2x-1}dx=\int e^{(2+\ln 5)x+\ln5-1}dx=5e^{-1}\int e^{(2+\ln 5)x}dx$$ Can you continue?

Use integration by parts. Set $2x=y$ and you'll get: $$5e^{-1}\int5^{\frac{y}{2}}e^ydy=5e^{-1}\int5^{\frac{y}{2}}d(e^y)$$ After that just a little effort. :) $$5e^{-1}\int5^{\frac{y}{2}}e^ydy=5e^{-1}\int5^{\frac{y}{2}}d(e^y)=5e^{-1}(5^{\frac{y}{2}}e^y-\frac{\ln(5)}{2}\int5^{\frac{y}{2}}e^ydy)$$ Combining the first and the last parts: $$5e^{-1}\int5^{\frac{y}{2}}e^ydy=5e^{-1}(5^{\frac{y}{2}}e^y-\frac{\ln(5)}{2}\int5^{\frac{y}{2}}e^ydy)$$ Cancelling out multipliers and collecting the integral terms:

$$(1+\frac{\ln(5)}{2})\int5^{\frac{y}{2}}e^ydy=5^{\frac{y}{2}}e^y$$ So one shoulf get: $$\int5^{\frac{y}{2}}e^ydy=\frac{5^{\frac{y}{2}}e^y}{1+\frac{\ln(5)}{2}}$$ So after that just remember the multiplier $5e^{-1}$ and the original variable $2x=y$

$$\displaystyle \int 5^{x+1}e^{2x-1}dx$$ $$=\int e^{(x+1)ln(5)}e^{2x-1}dx$$ $$=\int e^{(x+1)ln(5)+2x-1}dx$$ $$=\int e^{(\ln(5)+2)x+\ln(5)-1}dx$$ $$=\int e^{(\ln(5)+2)x}\cdot e^{\ln(5)-1}dx$$ $$=\frac{5}{e}\int e^{(\ln(5)+2)x}dx$$ $$=\frac{5}{e}\int e^{(\ln(5)+2)x}dx$$ $$=\frac{5^{x} e^{2 x}}{e \log{\left (5 \right )} + 2 e}$$