# Quick Integration by Parts Question

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!

• $\frac{2}{3} x^3$? – Adam Rubinson Oct 27 '20 at 1:52
• Ok, I'm a bit confused on how you got there though. – Tyler Oct 27 '20 at 1:57
• I would differentiate both sides and compare. – Kenta S Oct 27 '20 at 2:00
• Ok, so I'm still a little confused, if I take the derivative of an integral that cancels it out right? So it'd basically be the same thing as just taking the integral signs away and going from there? – Tyler Oct 27 '20 at 2:03
• IBP is: $\int u v’ dx = u v - \int u’ v dx$. Here, my v’ is $e^{\frac{\pi}{4}}$ and then I figured out that $u$ is $g(x)$ and must equal $\frac{2}{3} x^3$. – Adam Rubinson Oct 27 '20 at 2:03

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 $$g^{}(x)$$.