Respuesta :

Answer:convergent

Step-by-step explanation:

Given

Improper Integral I is given as

[tex]I=\int_{-\infty}^{0}1000e^xdx[/tex]

[tex]I=1000=\int_{-\infty}^{0}e^xdx[/tex]

integration of [tex]e^x[/tex] is [tex]e^x[/tex]

[tex]I=1000\times \left [ e^x\right ]^{0}_{-\infty}[/tex]

[tex]I=1000\times I=\left [ e^0-e^{-\infty}\right ][/tex]

[tex]I=1000\times \left [ e^0-\frac{1}{e^{\infty}}\right ][/tex]

[tex]I=1000\times 1=1000[/tex]

so the integration converges to 1000 units  

Space

Answer:

The improper integral converges.

[tex]\displaystyle \int\limits^0_{- \infty} {1000e^x} \, dx = 1000[/tex]

General Formulas and Concepts:
Calculus

Limits

Limit Rule [Variable Direct Substitution]:                                                         [tex]\displaystyle \lim_{x \to c} x = c[/tex]

Differentiation

  • Derivatives
  • Derivative Notation

Derivative Property [Multiplied Constant]:                                                       [tex]\displaystyle \frac{d}{dx} [cf(x)] = c \cdot f'(x)[/tex]  

Derivative Rule [Basic Power Rule]:

  1. f(x) = cxⁿ
  2. f’(x) = c·nxⁿ⁻¹

Integration

  • Integrals

Integration Rule [Reverse Power Rule]:                                                           [tex]\displaystyle \int {x^n} \, dx = \frac{x^{n + 1}}{n + 1} + C[/tex]

Integration Rule [Fundamental Theorem of Calculus 1]:                                 [tex]\displaystyle \int\limits^b_a {f(x)} \, dx = F(b) - F(a)[/tex]

Integration Property [Multiplied Constant]:                                                     [tex]\displaystyle \int {cf(x)} \, dx = c \int {f(x)} \, dx[/tex]

Integration Property [Addition/Subtraction]:                                                   [tex]\displaystyle \int {[f(x) \pm g(x)]} \, dx = \int {f(x)} \, dx \pm \int {g(x)} \, dx[/tex]

Integration Method: U-Substitution

Improper Integrals:                                                                                           [tex]\displaystyle \int\limits^{\infty}_a {f(x)} \, dx = \lim_{b \to \infty} \int\limits^b_a {f(x)} \, dx[/tex]

Step-by-step explanation:

Step 1: Define

Identify.

[tex]\displaystyle \int\limits^0_{- \infty} {1000e^x} \, dx[/tex]

Step 2: Integrate

  1. [Integral] Rewrite [Integration Property - Multiplied Constant]:             [tex]\displaystyle \int\limits^0_{- \infty} {1000e^x} \, dx = 1000 \int\limits^0_{- \infty} {e^x} \, dx[/tex]
  2. [Integral] Rewrite [Improper Integral]:                                                     [tex]\displaystyle \int\limits^0_{- \infty} {1000e^x} \, dx = \lim_{a \to -\infty} 1000 \int\limits^0_{a} {e^x} \, dx[/tex]
  3. [Integral] Apply Exponential Integration:                                                 [tex]\displaystyle \int\limits^0_{- \infty} {1000e^x} \, dx = \lim_{a \to -\infty} 1000 \big( e^x \big) \bigg| \limits^0_{a}[/tex]
  4. Evaluate [Integration Rule - Fundamental Theorem of Calculus]:         [tex]\displaystyle \int\limits^0_{- \infty} {1000e^x} \, dx = \lim_{a \to -\infty} 1000 \big( 1 - e^a \big)[/tex]
  5. [Limit] Evaluate [Limit Rule - Variable Direct Substitution]:                     [tex]\displaystyle \int\limits^0_{- \infty} {1000e^x} \, dx = 1000 \big( 1 - e^{- \infty} \big)[/tex]
  6. Rewrite:                                                                                                     [tex]\displaystyle \int\limits^0_{- \infty} {1000e^x} \, dx = 1000 \big( 1 - \frac{1}{e^{\infty}} \big)[/tex]
  7. Simplify:                                                                                                     [tex]\displaystyle \int\limits^0_{- \infty} {1000e^x} \, dx = 1000[/tex]

∴ the improper integral is equal to 1000 and is convergent.

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Learn more about improper integrals: https://brainly.com/question/14411888

Learn more about calculus: brainly.com/question/23558817

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Topic: AP Calculus BC (Calculus I + II)

Unit: Integration

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