Improper integrals of type 1

Improper integrals: Oneigenlijke integralen *

Improper integrals of type 1

Let $f(x)$ be a given integrand and $F(x)$ an antiderivative of $f(x)$ . We examine three cases of integration areas with infinite length.

If $f(x)$ is continuous on the interval $[a,\infty)$ and if $\displaystyle \lim_{N\to\infty} F(N)$ exists, then $\int_a^{\infty} f(x)\,\dd x = \lim_{N\to\infty} \int_a^{N} f(x)\,\dd x = \lim_{N\to\infty} F(N) - F(a)\tiny.$

If $f(x)$ is continuous on the interval $(-\infty,b]$ and if $\displaystyle \lim_{M\to -\infty} F(M)$ exists, then $\int_{-\infty}^b f(x)\,\dd x = \lim_{M\to -\infty} \int_M^b f(x)\,\dd x = F(b) - \lim_{M\to -\infty} F(M)\tiny.$

If $f(x)$ is contiuous on the interval $(-\infty,\infty)$ and if the limits $\displaystyle \lim_{N\to \infty} F(N)$ and $\displaystyle \lim_{M\to -\infty} F(M)$ exist and at least one of them is finite, then $\int_{-\infty}^{\infty} f(x)\,\dd x = \lim_{M\to-\infty} \lim_{N\to \infty} \int_M^{N} f(x)\,\dd x = \lim_{N\to \infty} F(N) - \lim_{M\to -\infty} F(M)\tiny.$

We present some illustrative examples.

Calculate the following improper integral or show that it diverges: $\int_{1}^{\infty}\frac{1}{x}\,\dd x$

$\displaystyle\int_{1}^{\infty}\frac{1}{x}\,\dd x={}$ $\infty$

$\int_1^{\infty}\frac{1}{x}\,\dd x=\infty$ want $\displaystyle\int_1^N\frac{1}{x}\,\dd x=\bigl[\ln(x)\bigr]_1^N=\ln(N)$ and $\displaystyle \lim_{N\to\infty}\ln(N)=\infty$ .

New example