I having trouble using partial fractions.

Question

For the integral below I have to use partial fractions, however I am at a lost on how to do so. $$\int\frac{dt}{t^2-t-20}$$

The farthest I have gotten to is factoring the denominator to $(t+5)(t-4)$. Thanks in advance.

Answer 1

Note that the signs you wrote are wrong.

$$\frac{1}{t^2-t-20}=\frac{1}{(t\color{red}{-}5)(t\color{red}{+}4)}=\frac 19\left(\frac{1}{t-5}-\frac{1}{t+4}\right)$$

Answer 2

hint: $\dfrac{1}{t^2-t-20} = -\dfrac{1}{9}\cdot \dfrac{1}{t+4} + \dfrac{1}{9}\cdot \dfrac{1}{t-5}$

Answer 3

When you have to solve something of the form:

$$\int \dfrac{dt}{(t-a)(t-b)} $$

Then you can have

$$\dfrac{A}{t-a} + \dfrac{B}{t-b} = \dfrac{1}{(t-a)(t-b)}$$

Wich means that

$$\left\{\begin{array}{rcl}A\ +B\ & = & 0 \\ Ab + Ba & = & -1 \end{array}\right. $$

Once you find $A$ and $B$ then

$$\int \dfrac{dt}{(t-a)(t-b)} = \int{\dfrac{A}{t-a}dt} + \int{\dfrac{B}{t-b}dt}$$