# Thread: concepts of improper integrals

1. ## concepts of improper integrals

$\displaystyle \int \frac{10}{x^2-2x}dx$ [0,3]

To determine the convergence or divergence of the integral, how many improper integrals must be analyzed? What must be true of each of these integrals if the given integral converges?

I think it must be analyzed twice. Even if that is right, I don't know what must be true of each of them if they converge.

2. Originally Posted by saiyanmx89
$\displaystyle \int \frac{10}{x^2-2x}dx$ [0,3]

To determine the convergence or divergence of the integral, how many improper integrals must be analyzed? What must be true of each of these integrals if the given integral converges?

I think it must be analyzed twice. Even if that is right, I don't know what must be true of each of them if they converge.

note that $\displaystyle \int_0^3 \frac{10}{x^2-2x}dx$ has to be broken up into three improper integral "pieces"

$\displaystyle \lim_{a \to 0} \int_a^1 \frac{10}{x(x-2)} \, dx$

$\displaystyle \lim_{b \to 2} \int_1^b \frac{10}{x(x-2)} \, dx$

$\displaystyle \lim_{b \to 2} \int_b^3 \frac{10}{x(x-2)} \, dx$

If the overall integral converges, then each "piece" of the integral must converge. if any single "piece" diverges, then the entire integral diverges.

3. Originally Posted by saiyanmx89
$\displaystyle \int \frac{10}{x^2-2x}dx$ [0,3]

To determine the convergence or divergence of the integral, how many improper integrals must be analyzed? What must be true of each of these integrals if the given integral converges?

I think it must be analyzed twice. Even if that is right, I don't know what must be true of each of them if they converge.

The integrand has infinite discontinuities at $\displaystyle x=0$ and $\displaystyle x=2.$ If the integral does converge, then we may write it as
$\displaystyle \int_0^3\frac{10}{x^2-2x}\,dx$
$\displaystyle =\int_0^1\frac{10}{x^2-2x}\,dx+\int_1^2\frac{10}{x^2-2x}\,dx+\int_2^3\frac{10}{x^2-2x}\,dx$ (I chose to split the first interval at 1, but you could use any value between 0 and 2)
$\displaystyle =\lim_{a\to0^+}\left[\int_a^1\frac{10}{x^2-2x}\,dx\right]+\lim_{b\to2^-}\left[\int_1^b\frac{10}{x^2-2x}\,dx\right]+\lim_{a\to2^+}\left[\int_a^3\frac{10}{x^2-2x}\,dx\right]$