Evaluate the integral.
Okay I'm going to have to assume you know how to deal with expression in the form of $\displaystyle \int_{}^{}{ae^{bx}}dx$ If you don't just say.
Now to integrate $\displaystyle 3^x$ we just need to rewrite it as e raised to an particular power, consider $\displaystyle y = 3^x$ then $\displaystyle ln(y) = ln(3^x) .... ln(y) = xln(3)$ now we just antilog both sides so $\displaystyle y = e^{xln3}$
so you understand how to write $\displaystyle 3^x$ as $\displaystyle e^{xln3}$
so to integrate you just use the chain rule backwards $\displaystyle \int_{}^{}{e^{xln3}}dx = \frac{e^{xln3}}{ln3} = \frac{3^{x}}{ln3} $
now in general $\displaystyle \int{a^x} dx = \frac{a^x}{lna}$
$\displaystyle (a^x)'=(e^{x\ln a})'=a^x\ln a.$
Integrate both sides
$\displaystyle a^x+k_1=\int a^x\ln a\,dx\,\therefore\,\frac{a^x}{\ln a}+k=\int a^x\,dx.$
So $\displaystyle \int_1^33^x\,dx=\left.\frac{3^x}{\ln3}\right|_1^3$
What's the problem with this?