Question

Hello there, I would like to ask a few questions.


1) 1/((x+1)(x^2+1)) can you please tell me what are the partial fractions of this fraction

2) can you please tell me what do we do in the case that we have to integrate an improper integral which does't have infinity (say a definite integral from 1 to 4)

I am asking these questions to see whether or not I have written the correct thing in my exams or not

Answer 1

\[\frac{ 1 }{ (x+1)(x^2+1) }\]
Are you trying to expand this using partial fractions?

Answer 2

@Christos

Answer 3

The first thing to do is see if we can factor it further, can we factor x^2+1 ?
or is that as simple as it gets?

Answer 4

Its as simple as it gets, right?

\[\frac{ 1 }{ (x+1)(x^2+1) } = \space \frac{ A }{ x+1 \space } \space + \space \frac{ cx +D }{ x^2+1 }\]

so for linear factors like x+1 we just put a constant on top, but for quadratics, we put a linear term on top and try and find A,C and D

Answer 5

And for your 2nd question, well, it depends what the integrate is...did you have that information? what function are we integrating?

Answer 6

\(\int_1^4 \frac{1}{x-1}dx\) (function goes to infinity as \(x\to1\)), would be computed as
\[
\lim_{t_0\stackrel{\ge}{\to}1} \int_{t_0}^4 \frac{1}{x-1} dx = \lim_{t_0 \stackrel{\ge}{\to}1} F(4)-F(t_0)
\]
where \(F\) is such that \(F'(x)=\frac{1}{x-1}\). For each \(t_0>1\),

Answer 7

\[
\lim_{t_0\stackrel{\ge}{\to}1} \ln|4-1| - \lim|t_0-1| = \ln3 - (-\infty) = +\infty
\]
In thsi example, the integral does not converge

Answer 8

any more