Respuesta :
Answer:
The improper integral converges.
[tex]\displaystyle \int\limits^{\infty}_0 {50e^{-50x}} \, dx = 1[/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]:
- f(x) = cxⁿ
- 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^{\infty}_0 {50e^{-50x}} \, dx[/tex]
Step 2: Integrate Pt. 1
- [Integral] Rewrite [Improper Integral]: [tex]\displaystyle \int\limits^{\infty}_0 {50e^{-50x}} \, dx = \lim_{b \to \infty} \int\limits^b_0 {50e^{-50x}} \, dx[/tex]
- [Integral] Rewrite [Integration Property - Multiplied Constant]: [tex]\displaystyle \int\limits^{\infty}_0 {50e^{-50x}} \, dx = \lim_{b \to \infty} 50 \int\limits^b_0 {e^{-50x}} \, dx[/tex]
Step 3: Integrate Pt. 2
Identify variables for u-substitution.
- Set u: [tex]\displaystyle u = -50x[/tex]
- [u] Differentiate [Derivative Properties and Rules]: [tex]\displaystyle du = -50 \ dx[/tex]
- [Bounds] Swap: [tex]\displaystyle \left \{ {{x = b \rightarrow u = -50b} \atop {x = 0 \rightarrow u = 0}} \right.[/tex]
Step 4: Integrate Pt. 3
- [Integral] Rewrite [Integration Property - Multiplied Constant]: [tex]\displaystyle \int\limits^{\infty}_0 {50e^{-50x}} \, dx = \lim_{b \to \infty} - \int\limits^b_0 {-50e^{-50x}} \, dx[/tex]
- [Integral] U-Substitution: [tex]\displaystyle \int\limits^{\infty}_0 {50e^{-50x}} \, dx = \lim_{b \to \infty} - \int\limits^{-50b}_0 {e^u} \, du[/tex]
- [Integral] Apply Exponential Integration: [tex]\displaystyle \int\limits^{\infty}_0 {50e^{-50x}} \, dx = \lim_{b \to \infty} -e^u \bigg| \limits^{-50b}_0[/tex]
- Evaluate [Integration Rule - Fundamental Theorem of Calculus 1]: [tex]\displaystyle \int\limits^{\infty}_0 {50e^{-50x}} \, dx = \lim_{b \to \infty} - \Big( e^{-50b} - 1 \Big)[/tex]
- Rewrite: [tex]\displaystyle \int\limits^{\infty}_0 {50e^{-50x}} \, dx = \lim_{b \to \infty} - \Big( \frac{1}{e^{50b}} - 1 \Big)[/tex]
- [Limit] Evaluate [Limit Rule - Variable Direct Substitution]: [tex]\displaystyle \int\limits^{\infty}_0 {50e^{-50x}} \, dx = - \Big( \frac{1}{e^{50(\infty)}} - 1 \Big)[/tex]
- Simplify: [tex]\displaystyle \int\limits^{\infty}_0 {50e^{-50x}} \, dx = 1[/tex]
∴ the improper integral equals to 1 and converges.
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Learn more about improper integrals: https://brainly.com/question/15086429
Learn more about calculus: https://brainly.com/question/23558817
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Topic: AP Calculus BC (Calculus I + II)
Unit: Integration
