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