## PROBLEM 3

Determine the primitives of the function $f: R \rightarrow R, f(x)=\frac{x(x-1)(x-2)}{\left(x^{2}-2 x\right)^{4}+4}$
Solution:

![](https://cdn.mathpix.com/cropped/2024_06_07_a4f09379d0be992cd7e1g-3.jpg?height=122&width=1559&top_left_y=2049&top_left_x=244)

Let $x^{2}-2 x=t \Rightarrow 2(x-1) d x=d t \Rightarrow(x-1) d x=\frac{d t}{2}$

Thus $F(x)=\int f(x) d x=\int \frac{t}{t^{4}+4} \cdot \frac{1}{2} d t$

Let $t^{2}=u \Rightarrow 2 t d t=d u \Rightarrow t d t=\frac{d u}{2}$

1 point

$$
\begin{aligned}
& F(u)=\frac{1}{4} \int \frac{1}{u^{2}+4} d u=\frac{1}{8} \operatorname{arctg} \frac{u}{2}+C \\
& 1 \text { point } \\
& F(x)=\frac{1}{8} \operatorname{arctg} \frac{\left(x^{2}-2 x\right)^{2}}{2}+C
\end{aligned}
$$