Write the suitable domain for $\int\frac{t^2}{t^2+5t+4}\,dt$

Question

$$\int\frac{t^2}{t^2+5t+4}\,dt$$

How can I write a domain? Is the domain $t\in\mathbb R\setminus\{0\}$? How can I write?

Answer 1

Hint: $$\frac{t^2}{t^2+5t+4} = \frac{1}{3(t+1)} - \frac{16}{3(t+4)} + 1$$

Answer 2

Just give your function a name like $F(t)\;(\,+C\,) = \int\frac{t^2}{t^2+5t+4}\,dt$. Here $C$ is the integration constant which has no influence on the domain. So you can ignore it.

Often the domain of a function with a name which is here $F$ is abbreviated symbolically by $D(F)$ or $domain(F)$ or $Dom(F)$.

Using the hint from GNU Supporter, you see that $F(t)$ is defined for all real $t$ except $t=-1$ and $t=-4$. So, you can write:

$D(F) = \mathbb{R}\setminus\{-1,-4\}$ or $D(F) = \{t \in \mathbb{R} | t\neq -1, t\neq -4 \}$ or $D(F) = \{t | t \in \mathbb{R}, t\neq -1, t\neq -4 \}$.

There are also other similar notations. Pick the one you like.