Question

How do you use partial fraction decomposition to decompose the fraction to integrate `(7)/(x^2+13x+40)`?

Answer 1

The integral equals `7/3ln|(x + 5)/(x +8)| + C`

Explanation:

We wish to find factors in the denominator. The trick is to find two numbers that multiply to `40` and add to `13`. Clearly these will be `8` and `5`.

`I = int 7/((x+ 5)(x+ 8))dx`

Now we can decompose in partial fractions.

`A/(x+ 5) + B/(x +8) = 7/((x +5)(x + 8))`

`A(x + 8) + B(x + 5) = 7`

`Ax + 8A + Bx + 5B = 7`

`(A + B)x + (8A + 5B) = 7`

Now we have a system of equations.

`{(A + B = 0), (8A + 5B = 7):}`

Substituting the first equation into the second we see that

`8A + 5(-A) = 7`

`3A = 7`

`A = 7/3`

Now clearly `B = -7/3` because `A+ B =0`. The integral becomes:

`I = int7/(3(x + 5)) - 7/(3(x + 8)) dx`

`I= 7/3ln|x +5| - 7/3ln|x + 8| + C`

`I = 7/3ln|(x + 5)/(x +8)| + C`

Hopefully this helps!