Calculate $\int 5^{x+1}e^{2x-1}\,dx$ How to calculate following integration?
$$\int 5^{x+1}e^{2x-1}dx$$

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 $2x=y$
