Improper Integrals

Part 1 of 7 — Infinite Limits of Integration

Type 1: Infinite Bounds

$int_a^{infty} f(x),dx = lim_{b o infty} int_a^b f(x),dx$

• If the limit exists → the integral converges
• If the limit is $pminfty$ or DNE → the integral diverges

Classic Example

Infinite Bounds 🎯

Key Takeaways — Part 1

1. Replace $\infty$ with a limit variable
2. Evaluate, then take the limit
3. $\int_1^{\infty} 1/x\,dx$ diverges but $\int_1^{\infty} 1/x^2\,dx$ converges

Improper Integrals

Part 2 of 7 — The $p$-Test

$p$-Integral Test

int_1^{infty} rac{1}{x^p},dx egin{cases} ext{converges} & ext{if } p > 1 \ ext{diverges} & ext{if } p leq 1 end{cases}

Key Examples

$p$-Test 🎯

Key Takeaways — Part 2

1. $p > 1$: converges. $p \leq 1$: diverges.
2. The boundary $p = 1$ ($\ln x$) is the dividing line

Improper Integrals

Part 3 of 7 — Discontinuous Integrands (Type 2)

Type 2: Vertical Asymptotes

If $f$ has a vertical asymptote at $x = c$ inside $[a, b]$:

$int_a^b f(x),dx = lim_{t o c^-}int_a^t f(x),dx + lim_{t o c^+}int_t^b f(x),dx$

Example

int_0^1 rac{1}{sqrt{x}},dx = lim_{t o 0^+}int_t^1 x^{-1/2},dx = lim_{t o 0^+}[2sqrt{x}]_t^1 = 2 - 0 = 2

Discontinuous Integrands 🎯

Key Takeaways — Part 3

1. Check for vertical asymptotes inside the interval
2. Split and use limits from the appropriate side

Improper Integrals

Part 4 of 7 — Comparison Test

Direct Comparison Test

For $0 leq f(x) leq g(x)$ on $[a, infty)$:

• If $int_a^{infty} g\,dx$ converges → $int_a^{infty} f\,dx$ converges
• If $int_a^{infty} f\,dx$ diverges → $int_a^{infty} g\,dx$ diverges

Bigger converges → smaller converges Smaller diverges → bigger diverges

Comparison Test 🎯

Key Takeaways — Part 4

Comparison test: bound by a known convergent/divergent integral.

Improper Integrals

Part 5 of 7 — Both-Sided Improper Integrals

Integrals from $-infty$ to $infty$

$int_{-infty}^{infty} f(x),dx = int_{-infty}^c f(x),dx + int_c^{infty} f(x),dx$

Both must converge independently!

Full Line Integrals 🎯

Key Takeaways — Part 5

Split at any point $c$ (usually 0) and evaluate each half.

Improper Integrals

Part 6 of 7 — Practice Workshop

Mixed Practice 🎯

Workshop Complete!

Improper Integrals — Review

Part 7 of 7 — Final Assessment

Final Assessment 🎯

Improper Integrals — Complete! ✅