Math 181 7.8 – Improper Integrals

−∞ 𝒃 𝒇 𝒙 𝒅𝒙 = 𝐥𝐢𝐦 𝒕→−∞ 𝒕 𝒃 𝒇 𝒙 𝒅𝒙 In general, these types of improper integrals (Type I) are defined like this: 𝒂 ∞ 𝒇 𝒙 𝒅𝒙 = 𝐥𝐢𝐦 𝒕→∞ 𝒂 𝒕 𝒇 𝒙 𝒅𝒙   −∞ 𝒃 𝒇 𝒙 𝒅𝒙 = 𝐥𝐢𝐦 𝒕→−∞ 𝒕 𝒃 𝒇 𝒙 𝒅𝒙 −∞ ∞ 𝒇 𝒙 𝒅𝒙 = −∞ 𝒄 𝒇 𝒙 𝒅𝒙 + 𝒄 ∞ 𝒇 𝒙 𝒅𝒙

−∞ 𝒃 𝒇 𝒙 𝒅𝒙 = 𝐥𝐢𝐦 𝒕→−∞ 𝒕 𝒃 𝒇 𝒙 𝒅𝒙 In general, these types of improper integrals (Type I) are defined like this: 𝒂 ∞ 𝒇 𝒙 𝒅𝒙 = 𝐥𝐢𝐦 𝒕→∞ 𝒂 𝒕 𝒇 𝒙 𝒅𝒙   −∞ 𝒃 𝒇 𝒙 𝒅𝒙 = 𝐥𝐢𝐦 𝒕→−∞ 𝒕 𝒃 𝒇 𝒙 𝒅𝒙 −∞ ∞ 𝒇 𝒙 𝒅𝒙 = −∞ 𝒄 𝒇 𝒙 𝒅𝒙 + 𝒄 ∞ 𝒇 𝒙 𝒅𝒙

−∞ 𝒃 𝒇 𝒙 𝒅𝒙 = 𝐥𝐢𝐦 𝒕→−∞ 𝒕 𝒃 𝒇 𝒙 𝒅𝒙 In general, these types of improper integrals (Type I) are defined like this: 𝒂 ∞ 𝒇 𝒙 𝒅𝒙 = 𝐥𝐢𝐦 𝒕→∞ 𝒂 𝒕 𝒇 𝒙 𝒅𝒙   −∞ 𝒃 𝒇 𝒙 𝒅𝒙 = 𝐥𝐢𝐦 𝒕→−∞ 𝒕 𝒃 𝒇 𝒙 𝒅𝒙 −∞ ∞ 𝒇 𝒙 𝒅𝒙 = −∞ 𝒄 𝒇 𝒙 𝒅𝒙 + 𝒄 ∞ 𝒇 𝒙 𝒅𝒙

In each case above, if the limit is finite, we say the improper integral __________. If the limit does not exist, the improper integral __________.

In each case above, if the limit is finite, we say the improper integral __________. If the limit does not exist, the improper integral __________. converges

In each case above, if the limit is finite, we say the improper integral __________. If the limit does not exist, the improper integral __________. converges diverges

Ex 1. Does 1 ∞ 1 𝑥 𝑑𝑥 converge or diverge?

Ex 2. Evaluate: −∞ ∞ 𝑑𝑥 1+ 𝑥 2

Ex 3. For what values of 𝑝 does the integral 1 ∞ 𝑑𝑥/ 𝑥 𝑝 converge Ex 3. For what values of 𝑝 does the integral 1 ∞ 𝑑𝑥/ 𝑥 𝑝 converge? When the integral does converge, what is its value?

1 ∞ 1 𝑥 𝑝 𝑑𝑥 … …converges to 1 𝑝−1 if 𝑝>1. …diverges if 𝑝≤1.

In general, these types of improper integrals (Type II) are defined like this: If 𝑓(𝑥) continuous on 𝑎,𝑏 and discontinuous at 𝑎, then 𝒂 𝒃 𝒇 𝒙 𝒅𝒙 = 𝐥𝐢𝐦 𝒕→ 𝒂 + 𝒕 𝒃 𝒇 𝒙 𝒅𝒙 If 𝑓(𝑥) continuous on 𝑎,𝑏 and discontinuous at 𝑏, then 𝒂 𝒃 𝒇 𝒙 𝒅𝒙 = 𝐥𝐢𝐦 𝒕→ 𝒃 − 𝒂 𝒕 𝒇 𝒙 𝒅𝒙 If 𝑓 𝑥 discontinuous at 𝑐, where 𝑎<𝑐<𝑏, and continuous on 𝑎,𝑐 ∪ 𝑐,𝑏 , then 𝒂 𝒃 𝒇 𝒙 𝒅𝒙 = 𝒂 𝒄 𝒇(𝒙) 𝒅𝒙+ 𝒄 𝒃 𝒇(𝒙) 𝒅𝒙

In general, these types of improper integrals (Type II) are defined like this: If 𝑓(𝑥) continuous on 𝑎,𝑏 and discontinuous at 𝑎, then 𝒂 𝒃 𝒇 𝒙 𝒅𝒙 = 𝐥𝐢𝐦 𝒕→ 𝒂 + 𝒕 𝒃 𝒇 𝒙 𝒅𝒙 If 𝑓(𝑥) continuous on 𝑎,𝑏 and discontinuous at 𝑏, then 𝒂 𝒃 𝒇 𝒙 𝒅𝒙 = 𝐥𝐢𝐦 𝒕→ 𝒃 − 𝒂 𝒕 𝒇 𝒙 𝒅𝒙 If 𝑓 𝑥 discontinuous at 𝑐, where 𝑎<𝑐<𝑏, and continuous on 𝑎,𝑐 ∪ 𝑐,𝑏 , then 𝒂 𝒃 𝒇 𝒙 𝒅𝒙 = 𝒂 𝒄 𝒇(𝒙) 𝒅𝒙+ 𝒄 𝒃 𝒇(𝒙) 𝒅𝒙

In general, these types of improper integrals (Type II) are defined like this: If 𝑓(𝑥) continuous on 𝑎,𝑏 and discontinuous at 𝑎, then 𝒂 𝒃 𝒇 𝒙 𝒅𝒙 = 𝐥𝐢𝐦 𝒕→ 𝒂 + 𝒕 𝒃 𝒇 𝒙 𝒅𝒙 If 𝑓(𝑥) continuous on 𝑎,𝑏 and discontinuous at 𝑏, then 𝒂 𝒃 𝒇 𝒙 𝒅𝒙 = 𝐥𝐢𝐦 𝒕→ 𝒃 − 𝒂 𝒕 𝒇 𝒙 𝒅𝒙 If 𝑓 𝑥 discontinuous at 𝑐, where 𝑎<𝑐<𝑏, and continuous on 𝑎,𝑐 ∪ 𝑐,𝑏 , then 𝒂 𝒃 𝒇 𝒙 𝒅𝒙 = 𝒂 𝒄 𝒇(𝒙) 𝒅𝒙+ 𝒄 𝒃 𝒇(𝒙) 𝒅𝒙

In each case above, if the limit is finite, we say the improper integral __________. If the limit does not exist, the improper integral __________.

In each case above, if the limit is finite, we say the improper integral __________. If the limit does not exist, the improper integral __________. converges

In each case above, if the limit is finite, we say the improper integral __________. If the limit does not exist, the improper integral __________. converges diverges

Ex 4. Evaluate: 0 1 1 1−𝑥 𝑑𝑥

Ex 5. Evaluate: 0 3 1 𝑥−1 2/3 𝑑𝑥

To determine if an integral converges or diverges, you can use the Direct Comparison Test, or the Limit Comparison Test, which are described below.

Comparison Test Suppose 𝑓 and 𝑔 are continuous on 𝑎,∞ and 0≤𝑓 𝑥 ≤𝑔(𝑥) in 𝑎,∞ . If 𝑎 ∞ 𝑔(𝑥) 𝑑𝑥 converges, then 𝑎 ∞ 𝑓(𝑥) 𝑑𝑥 converges. If 𝑎 ∞ 𝑓(𝑥) 𝑑𝑥 diverges, then 𝑎 ∞ 𝑔(𝑥) 𝑑𝑥 diverges.

Ex 6. Does 1 ∞ sin 2 𝑥 𝑥 2 𝑑𝑥 converge or diverge?

Ex 7. Does 1 ∞ 1 𝑥 2 −0.1 𝑑𝑥 converge or diverge?