Q:

Doing a Line Integral Problem

Here is my attempt:
$$W=\int_C\vec{F}\cdot d\vec{r}\\=\int_C\frac{\alpha x}{(x^2+y^2)^{3/2}}dx+\frac{\alpha y}{(x^2+y^2)^{3/2}}dy\\Using\quad x=2t+1\quad and \quad y=-2t\quad for\quad 0\le t\le 1\\=\alpha\int_0^1\frac{8t+2}{[(2t+1)^2+4t^2]^{3/2}}dt$$
At this point, I am stuck.

A:

Hints:
$$\begin{align*}\bullet&\;\;(2t+1)^2+4t^2=8t^2+4t+1\\{}\\
\bullet&\;\;8t+2=\frac12\left(8t^2+4t+1\right)'\\{}\\
\bullet&\;\;\forall\,\text{differentiable positive function}\;f(x)\;,\;\;\int\frac{f'(x)\,dx}{f(x)^{3/2}}=-\frac 2{\sqrt{f(x)}}+C\end{align*}$$

