So, came across the following question while I was studying for a test:
"Let $g$ be a differentiable function such that $\int g(x)e^{\frac{x}{4}}dx=4g(x)e^{\frac{x}{4}}-\int 8x^2e^{\frac{x}{4}}dx$ What's a possible expression for $g(x)$?"
We're talking about integration by parts, and I assumed that I should apply that formula ($\int udv = uv-\int vdu$) here, but it seems like after using that that there's no formula given for $g(x)$? I'm a bit lost and any help would be appreciated!
I write down the two terms so that you can compare them.
$$\int g(x)\cdot e^{\frac{x}{4}}\, dx=g(x)\cdot 4e^{\frac{x}{4}}-\int 8x^2e^{\frac{x}{4}}\, dx$$
$\begin{array}{} \qquad \qquad \qquad \qquad \qquad \quad \Huge\Updownarrow \large ➀ & \qquad \Huge\Updownarrow \large ➁ & \quad \ \Huge\Updownarrow \large ➂ \end{array}$
$$\int g(x)^{}\cdot h^{'}(x)\, dx=g(x)\cdot h^{}(x)-\int g(x)^{'}\cdot h^{}(x)\, dx$$
$\large ➀$ Identify $h^{'}(x)$
$\large ➁$ Check if $h^{}(x)=4e^{\frac{x}4}$
$\large ➂$ Extract $h(x)$ so that the remaining part is $g^{'}(x)$. Then integrate to obtain $g^{}(x)$.