Expii

# Computing Improper Integrals with an Endpoint at Infinity - Expii

An improper integral with a single endpoint at infinity, like $$\int_a^\infty f(x)\,dx$$ or $$\int_{-\infty}^b f(x)\,dx$$, can be interpreted as a single limit of definite integrals, as long as $$f$$ has no infinite singularities on the interval of integration (namely, $$[a,\infty)$$ or $$(-\infty,b]$$). Specifically, $$\int_a^\infty f(x)\,dx$$ is defined as $$\lim_{t\to\infty} \int_a^t f(x)\,dx$$, while $$\int_{-\infty}^b f(x)\,dx$$ is defined as $$\lim_{t\to-\infty} \int_t^b f(x)\,dx$$.