$I=\displaystyle \int^\var{i1}_\var{i0} \simplify[std]{({a}x)*e^({c}x)} dx$

The formula for integration by parts is

$$\int u\frac{dv}{dx} dx = uv - \int v \frac{du}{dx} dx.$$

What is the most suitable choice for $u$ and $\frac{dv}{dx}$?

$u =\;$interpreted as

$\displaystyle{}$

Expected answer:interpreted as

$\displaystyle{3 x}$

$\frac{dv}{dx} =\;$interpreted as

$\displaystyle{}$

Expected answer:interpreted as

$\displaystyle{e^{ 4 x }}$

Hence find $\frac{du}{dx} =\;$interpreted as

$\displaystyle{}$

Expected answer:interpreted as

$\displaystyle{3}$

$v =\;$interpreted as

$\displaystyle{}$

Expected answer:interpreted as

$\displaystyle{\frac{ 1 }{ 4 } e^{ 4 x }}$

Hence find $uv =\;$interpreted as

$\displaystyle{}$

Expected answer:interpreted as

$\displaystyle{\frac{ 3 x }{ 4 } e^{ 4 x }}$

$\int v\frac{du}{dx}\mathrm{d}x = \;$interpreted as

$\displaystyle{}$

Expected answer:interpreted as

$\displaystyle{\frac{ 3 }{ 16 } e^{ 4 x }}$

Use the results from above to find:

$I=\displaystyle \int^\var{i1}_\var{i0} \simplify[std]{({a}x)*e^({c}x)} dx = \int^\var{i1}_\var{i0} u\frac{dv}{dx} dx = [uv]^\var{i1}_\var{i0} - \int^\var{i1}_\var{i0} v \frac{du}{dx} dx = \;$interpreted as

$\displaystyle{}$

Expected answer:interpreted as

$\displaystyle{\frac{ 163.7944500994 }{ 4 } - \frac{ 163.7944500994 }{ 16 } + \frac{ 3 }{ 16 }}$

Input all numbers as fractions or integers and not decimals.