Evaluating Improper Integrals - AP Calculus BC

1. Determine convergence: $\int_{0}^{\infty} e^{-x^2} \, dx$.

   Converges. Gaussian integral converges due to exponential decay.

2. When does an integral $\textstyle \int_{a}^{b} f(x) \, dx$ become improper?

   If $a$, $b$, or $f(x)$ are infinite or unbounded. Any infinite bound or unbounded integrand makes it improper.

3. Evaluate: \textstyle \text{∫}_{0}^{1} \frac{1}{\text{√}x} \text{dx}.

   Converges to $2$. $\lim_{a \to 0^+} [2\sqrt{x}]_a^1 = 2\sqrt{1} - 0 = 2$.

4. What does it mean if an improper integral converges?

   The limit defining the integral exists and is finite. The limit exists and equals a finite real number.

5. Identify convergence: $\int_{1}^{\infty} \frac{1}{\sqrt{x}} \, dx$.

   Diverges. Since $p = \frac{1}{2} \leq 1$, the integral diverges.

6. Identify the singularity: $\textstyle \int_{0}^{1} \frac{1}{x} dx$.

   Unbounded integrand at $x = 0$. Integrand has vertical asymptote at the lower limit.

7. Determine behavior: \textstyle \text{∫}_{0}^{1} \frac{1}{\text{√}x} \text{dx}

   Converges. Power $-\frac{1}{2} > -1$ ensures convergence at $x = 0$.

8. Evaluate: \textstyle \text{∫}_{0}^{\text{∞}} e^{-x} \text{dx}.

   Converges to $1$. $\lim_{b \to \infty} [-e^{-x}]_0^b = 0 - (-1) = 1$.

9. Convert to limit form: $\textstyle ∫_{-∞}^{0} e^{x} \, dx$.

   $\textstyle \lim_{a \to -∞} ∫_{a}^{0} e^{x} \, dx$. Negative infinite lower limit requires limit form.

10. Convert to limit: \textstyle \text{∫}_{0}^{1} \frac{1}{x} \text{dx}.

    \textstyle \text{lim}_{a \to 0^+} \text{∫}_{a}^{1} \frac{1}{x} \text{dx}. Integrand $\frac{1}{x}$ is unbounded at lower limit $x = 0$.

11. Identify singularity: $\int_{1}^{\infty} \frac{1}{(x-1)^2} dx$.

    Unbounded integrand at $x=1$. Function becomes infinite at the lower limit $x = 1$.

12. Evaluate: $\textstyle \int_{1}^{\infty} e^{-x} \text{dx}$

    Converges to $\frac{1}{e}$. $\lim_{b \to \infty} [-e^{-x}]_1^b = 0 - (-e^{-1}) = \frac{1}{e}$

13. Evaluate: $\int_{0}^{1} \frac{1}{\sqrt{1-x^2}} \, dx$.

    Converges to $\frac{\pi}{2}$. Arcsine function evaluation: $\arcsin(1) - \arcsin(0) = \frac{\pi}{2}$.

14. State the condition for the convergence of $\int_{0}^{a} (a-x)^{-1/2} \, dx$.

    Converges if $a > 0$. For $a > 0$, the singularity at $x = a$ is integrable.

15. What is convergence for \textstyle \text{∫}_{1}^{\text{∞}} \frac{1}{x^4} \text{dx}?

    Converges to $\frac{1}{3}$. $\lim_{b \to \infty} [-\frac{1}{3x^3}]_1^b = 0 - (-\frac{1}{3}) = \frac{1}{3}$.

16. What is an improper integral?

    An integral with infinite limits or an unbounded integrand. Defined when limits of integration or integrand are infinite.

17. What is convergence for $\int_1^\infty \frac{1}{x^4} \, dx$?

    Converges to $\frac{1}{3}$. $\lim_{b \to \infty} [-\frac{1}{3x^3}]_1^b = 0 - (-\frac{1}{3}) = \frac{1}{3}$.

18. Evaluate limit: $\textstyle \lim_{a \to 0^+} \int_{a}^{1} \frac{1}{x} dx$.

    Diverges. $\lim_{a \to 0^+} [\ln x]_a^1 = 0 - (-\infty) = \infty$.

19. Determine behavior: $\textstyle \int_{0}^{1} \frac{1}{x^p} \text{dx}$ for $p<1$.

    Converges. When $p < 1$, the singularity at $x = 0$ is integrable.

20. Find the limit: $\textstyle \lim_{b \to \infty} \int_{1}^{b} \frac{1}{x^3} dx$.

    Converges to $\frac{1}{2}$. $\lim_{b \to \infty} [-\frac{1}{2x^2}]_1^b = 0 - (-\frac{1}{2}) = \frac{1}{2}$.

21. Evaluate: $\int_{1}^{\infty} e^{-x} \, dx$.

    Converges to $\frac{1}{e}$. $\lim_{b \to \infty} [-e^{-x}]_1^b = 0 - (-e^{-1}) = \frac{1}{e}$

22. Find the limit: $\textstyle \lim_{b \to \infty} \int_{1}^{b} \frac{1}{x^3} \text{dx}$

    Converges to $\frac{1}{2}$. $\lim_{b \to \infty} [-\frac{1}{2x^2}]_1^b = 0 - (-\frac{1}{2}) = \frac{1}{2}$

23. Convert to limit: $\textstyle \int_{0}^{\infty} x e^{-x} \, dx$.

    $\textstyle \lim_{b \to \infty} \int_{0}^{b} x e^{-x} \, dx$. Infinite upper limit requires limit evaluation.

24. Evaluate: $\int_{1}^{\infty} \frac{1}{x^2} dx$

    Converges to $1$. $\lim_{b \to \infty} [-\frac{1}{x}]_1^b = \lim_{b \to \infty} (0 + 1) = 1$

25. $\textstyle \int_{0}^{\infty} x e^{-x} \, dx$

    $\textstyle \lim_{b \to \infty} \int_{0}^{b} x e^{-x} \, dx$. Infinite upper limit requires limit evaluation.

26. Determine convergence: $\textstyle \int_{0}^{\infty} e^{-x^2} \text{dx}$

    Converges. Gaussian integral converges due to exponential decay.

27. What is the behavior of \textstyle \text{∫}_{0}^{1} x^{-1/3} \text{dx}?

    Converges. Power $-\frac{1}{3} > -1$ ensures convergence.

28. Identify the type of improper integral: $\int_{0}^{1} \frac{1}{\sqrt{x}} \text{dx}$.

    Unbounded integrand at $x = 0$. $\frac{1}{\sqrt{x}}$ becomes infinite as $x \to 0^+.$

29. State the behavior of $\int_{0}^{1} x^{-1} dx$.

    Diverges. Same as $\int_0^1 \frac{1}{x} dx$ which diverges.

30. Identify singularity: $\textstyle \int_{1}^{\infty} \frac{1}{(x-1)^2} \text{d}x$.

    Unbounded integrand at $x=1$. Function becomes infinite at the lower limit $x = 1$.

31. Determine behavior: $\int_{0}^{1} \frac{1}{\sqrt{x}} \mathrm{d}x$.

    Converges. Power $-\frac{1}{2} > -1$ ensures convergence at $x = 0$.

32. Evaluate: $\textstyle \int_{1}^{\infty} \frac{1}{x (\ln x)^2} \, dx$.

    Converges. Higher power in denominator ensures convergence.

33. What is the behavior of $\int_{-\infty}^{\infty} e^{-x^2} \, dx$?

    Converges to $\sqrt{\pi}$. Famous Gaussian integral equals $\sqrt{\pi}$.

34. Identify convergence: $\int_{1}^{\infty} \frac{1}{\sqrt{x}} dx$.

    Diverges. Since $p = \frac{1}{2} \leq 1$, the integral diverges.

35. Convert to limit form: $\int_{-\infty}^{0} e^{x} \, dx$.

    $\lim_{a \to -\infty} \int_{a}^{0} e^{x} \, dx$. Negative infinite lower limit requires limit form.

36. When does an integral $\int_{a}^{b} f(x) \, dx$ become improper?

    If $a$, $b$, or $f(x)$ are infinite or unbounded. Any infinite bound or unbounded integrand makes it improper.

37. Evaluate limit: $\textstyle \lim_{a \to 0^+} \int_{a}^{1} \frac{1}{x} \text{dx}$

    Diverges. $\lim_{a \to 0^+} [\ln x]_a^1 = 0 - (-\infty) = \infty$

38. Identify the singularity: $\textstyle \int_{0}^{1} \frac{1}{x} \, dx$.

    Unbounded integrand at $x = 0$. Integrand has vertical asymptote at the lower limit.

39. What is the behavior of $\int_{-\infty}^{0} e^x \, dx$?

    Converges to $1$. $\lim_{a \to -\infty} [e^x]_a^0 = 1 - 0 = 1$.

40. Explain divergence in the context of improper integrals.

    The integral's limit is infinite or does not exist. The limit is infinite or undefined.

41. Convert to limit: $\int_{0}^{1} \frac{1}{x} \, \mathrm{d}x$.

    $\lim_{a \to 0^+} \int_{a}^{1} \frac{1}{x} \, \mathrm{d}x$. Integrand $\frac{1}{x}$ is unbounded at lower limit $x = 0$.

42. What is the behavior of \textstyle \text{∫}_{-\text{∞}}^{0} e^x \text{dx}?

    Converges to $1$. $\lim_{a \to -\infty} [e^x]_a^0 = 1 - 0 = 1$.

43. State the condition for the convergence of $\textstyle \int_{0}^{a} (a-x)^{-1/2} \, dx$.

    Converges if $a > 0$. For $a > 0$, the singularity at $x = a$ is integrable.

44. Determine behavior: $\textstyle \int_{0}^{1} \frac{1}{x^p} dx$ for $p<1$.

    Converges. When $p < 1$, the singularity at $x = 0$ is integrable.

45. Evaluate: $\textstyle \int_{0}^{1} \frac{1}{\sqrt{1-x^2}} \, dx$.

    Converges to $\frac{\pi}{2}$. Arcsine function evaluation: $\arcsin(1) - \arcsin(0) = \frac{\pi}{2}$.

46. Evaluate: \textstyle \text{∫}_{1}^{\text{∞}} \frac{1}{x^2} \text{dx}.

    Converges to $1$. $\lim_{b \to \infty} [-\frac{1}{x}]_1^b = \lim_{b \to \infty} (0 + 1) = 1$.

47. State the behavior of $\int_0^1 x^{-1} \text{dx}$

    Diverges. Same as $\int_0^1 \frac{1}{x} dx$ which diverges.

48. Determine convergence: $\textstyle \int_{1}^{\infty} \frac{1}{x} \text{dx}$.

    Diverges. $\ln(x)$ grows without bound, so the integral diverges.

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