🎯⭐ INTERACTIVE LESSON

Improper Integrals

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Improper Integrals - Complete Interactive Lesson

Part 1: Infinite Limits of Integration

Improper Integrals

Part 1 of 7 — Infinite Limits of Integration

Type 1: Infinite Bounds

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

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

Classic Example

$int_1^{infty} \frac{1}{x^2},dx = lim_{b \to infty} left[-\frac{1}{x}\right]_1^b = lim_{b \to infty}left(-\frac{1}{b} + 1\right) = 1$

Converges! The infinite area under $1/x^2$ is exactly $1$ .

Infinite Bounds 🎯

Key Takeaways — Part 1

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

Part 2: The p-Test

Improper Integrals

Part 2 of 7 — The $p$ -Test

$p$ -Integral Test

$int_1^{infty} \frac{1}{x^p},dx \begin{cases} \text{converges} & \text{if } p > 1 \ \text{diverges} & \text{if } p leq 1 end{cases}$

Key Examples

Integral	$p$	Result
$i n t_{1}^{i n f t y} 1$

Part 3: Discontinuous Integrands (Type 2)

Improper Integrals

Part 3 of 7 — Discontinuous Integrands (Type 2)

Type 2: Vertical Asymptotes

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

$i n t_{a}^{b} f (x), d$

Part 4: Comparison Test

Improper Integrals

Part 4 of 7 — Comparison Test

Direct Comparison Test

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

Part 5: Both-Sided Improper Integrals

Improper Integrals

Part 5 of 7 — Both-Sided Improper Integrals

Integrals from $-infty$ to $infty$

$i n t_{- i n f t y}^{i n f t y} f (x), d x = i n t_{- i n f t y}^{c} f (x),$

Part 6: Practice Workshop

Improper Integrals

Part 6 of 7 — Practice Workshop

Mixed Practice 🎯

Workshop Complete!

Part 7: Final Assessment

Improper Integrals — Review

Part 7 of 7 — Final Assessment

Final Assessment 🎯

Improper Integrals — Complete! ✅

\int_{1}^{\infty} 1/ x^{2} d x

in t_{1}^{in f t y} 1/ x^{2} d x

Key Takeaways — Part 2

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

in t_{a}^{b} f (x), d x = l i m_{t \to c^{-}} in t_{a}^{t} f (x), d x + l i m_{t \to c^{+}} in t_{t}^{b} f (x), d x

$int_0^1 \frac{1}{sqrt{x}},dx = lim_{t \to 0^+}int_t^1 x^{-1/2},dx = lim_{t \to 0^+}[2sqrt{x}]_t^1 = 2 - 0 = 2$

Discontinuous Integrands 🎯

Key Takeaways — Part 3

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

int_a^{infty} g\,dx

converges →

int_a^{infty} f\,dx

converges

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.

in t_{- in f t y}^{in f t y} f (x), d x = in t_{- in f t y}^{c} f (x), d x + in t_{c}^{in f t y} f (x), d x

Both must converge independently!

Full Line Integrals 🎯

Key Takeaways — Part 5

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

Improper Integrals

Improper Integrals - Complete Interactive Lesson

Part 1: Infinite Limits of Integration

Improper Integrals

Type 1: Infinite Bounds

Classic Example

Key Takeaways — Part 1

Part 2: The p-Test

Improper Integrals

ppp-Integral Test

Key Examples

Part 3: Discontinuous Integrands (Type 2)

Improper Integrals

Type 2: Vertical Asymptotes

Part 4: Comparison Test

Improper Integrals

Direct Comparison Test

Part 5: Both-Sided Improper Integrals

Improper Integrals

Integrals from −infty-infty−infty to inftyinftyinfty

Part 6: Practice Workshop

Improper Integrals

Workshop Complete!

Part 7: Final Assessment

Improper Integrals — Review

Improper Integrals — Complete! ✅

Key Takeaways — Part 2

Example

Key Takeaways — Part 3

Key Takeaways — Part 4

Key Takeaways — Part 5

$p$ -Integral Test

Integrals from $-infty$ to $infty$