Answer
$$\frac{10^x e^x}{1+\ln 10}+c$$
Work Step by Step
We do the integration by parts; we choose $u=10^x$ and $dv=e^xdx$. Then, $du=10^x\ln 10$, $v=e^x$
$$I=\int e^x 10^x dx=\int udv=uv-\int vdu\\
=10^x e^x- (\ln 10)\int e^x 10^x dx\\
=10^x e^x-I \ln 10 $$
Hence $$\int e^x 10^x dx=\frac{10^x e^x}{1+\ln 10}+c$$
