Why do we have to add a term for every exponent when integrating by partial fractons?

Question

For example, to integrate:

$1/(1+x)^3$ you can write it as a partial fraction as $A/(1+x) + B/(1+x)^2 + C/(1+x)^3$ but i don't really understand why.

I found that it is analogous to representing for example 7/16 as a/2 + b/4 + c/8 + d/16 but despite over an hour of googling i couldn't find a proof or reason as to why we have to use every term. Any help is highly appreciated

Answer 1

Basically when you write a partial fraction decomposition, you have to account for all possible factors of the original denominator. So all the possible factors of $(1+x)^3$ are $1+x$, $(1+x)^2$, and $(1+x)^3$ and then each of these possible factors become a term in the partial fraction decomposition.

To go back to your analogy, can you write $7/16$ as a fraction with just reduced fractions of the form $a/4$ and $b/8$? No, because then $4a+2b=7$ would have an integer solution which is impossible.

Answer 2

I have the same question, though I didn't understand the last sentence:

"Can you write 7/16 as a fraction with just reduced fractions of the form a/4 and b/8? No, because then 4a+2b=7 would have an integer solution which is impossible."

What do you mean that you would have an integral solution? If we use the analogy:

a/2 + b/4 + c/8 + d/16 = 7/16, we get 8a + 4b + 2c + d = 7. Why is that permissible while the 4a + 2b = 7 is incorrect?