Improper Integrals

Type I

Let $f (x)$ be continuous on $[𝒶, \infty)$ then

\int_{𝒶}^{\infty} f (x) ⅆ x = \lim_{𝒷 \to \infty} \int_{𝒶}^{𝒷} f (x) ⅆ x

If the limit $ℒ$ is finite then the improper integral converges and $ℒ$ is the value of the integral, otherwise the improper integral diverges.

Let $f (x)$ be continuous on $(-∞, 𝒷]$ then

\int_{-∞}^{𝒷} f (x) ⅆ x = \lim_{𝒶 \to -∞} \int_{𝒶}^{𝒷} f (x) ⅆ x

If the limit $ℒ$ is finite then the improper integral converges and $ℒ$ is the value of the integral, otherwise the improper integral diverges.

Let $f (x)$ be continuous on $(-∞, \infty)$ then

\int_{-∞}^{\infty} f (x) ⅆ x = \int_{-∞}^{𝒸} f (x) ⅆ x + \int_{𝒸}^{\infty} f (x) ⅆ x

where $𝒸$ is any real value.

Let $f (x)$ and $g (x)$ be continuous on $[𝒶, \infty)$ such that

\forall x \in [𝒶, \infty), 0 \leq f (x) \leq g (x)

then

If $\int_{𝒶}^{\infty} f (x) ⅆ x$ diverges then $\int_{𝒶}^{\infty} g (x) ⅆ x$ diverges.
If $\int_{𝒶}^{\infty} g (x) ⅆ x$ converges then $\int_{𝒶}^{\infty} f (x) ⅆ x$ converges.

Let $f (x)$ be continuous on $(𝒶, 𝒷]$ then

\int_{𝒶}^{𝒷} f (x) ⅆ x = \lim_{u \to 𝒶^{+}} \int_{u}^{𝒷} f (x) ⅆ x

If the limit $ℒ$ is finite then the improper integral converges and $ℒ$ is the value of the integral, otherwise the improper integral diverges.

Let $f (x)$ be continuous on $[𝒶, 𝒷)$ then

\int_{𝒶}^{𝒷} f (x) ⅆ x = \lim_{u \to 𝒷^{-}} \int_{𝒶}^{u} f (x) ⅆ x

If the limit $ℒ$ is finite then the improper integral converges and $ℒ$ is the value of the integral, otherwise the improper integral diverges.

Let $f (x)$ be continuous on $[𝒶, 𝒷]$ except at $𝒸 \in (𝒶, 𝒷)$ then

\int_{𝒶}^{𝒷} f (x) ⅆ x = \int_{𝒶}^{𝒸} f (x) ⅆ x + \int_{𝒸}^{𝒷} f (x) ⅆ x