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GaryK [48]
3 years ago
13

Https://files-cdn.schoology.com/eff1043385e13ac014eaf6e69f84525b?content-type=app

Mathematics
1 answer:
Anarel [89]3 years ago
8 0

Answer:

srry i need points

Step-by-step explanation:

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<h3>What are parallel lines and perpendicular lines ?</h3>

It is argued that two non-vertical lines in the same plane that have the same slope are parallel.

In terms of geometry, parallel lines are two separate lines that never cross each other and are located in the same plane. They may be vertical or horizontal. Examples of parallel lines can be found all around us on railroad tracks, in lines of notebooks, and at zebra crossings in daily life. It is impossible for two parallel lines to cross. Perpendicular lines are those that cross at a right angle when two non-vertical lines in the same plane.

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Anarel [89]

Answer:

\displaystyle \int\limits^1_0 {\frac{1}{xe^{x^2}}} \, dx = \infty

General Formulas and Concepts:

<u>Algebra I</u>

  • Exponential Rule [Rewrite]:                                                                           \displaystyle b^{-m} = \frac{1}{b^m}

<u>Calculus</u>

Limits

  • Right-Side Limit:                                                                                             \displaystyle  \lim_{x \to c^+} f(x)

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

Derivatives

Derivative Notation

Basic Power Rule:

  • f(x) = cxⁿ
  • f’(x) = c·nxⁿ⁻¹

Integrals

  • Definite Integrals

Integration Constant C

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

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

U-Substitution

U-Solve

Improper Integrals

Exponential Integral Function:                                                                              \displaystyle \int {\frac{e^x}{x}} \, dx = Ei(x) + C

Step-by-step explanation:

<u>Step 1: Define</u>

<em>Identify</em>

\displaystyle \int\limits^1_0 {\frac{1}{xe^{x^2}} \, dx

<u>Step 2: Integrate Pt. 1</u>

  1. [Integral] Rewrite [Exponential Rule - Rewrite]:                                          \displaystyle \int\limits^1_0 {\frac{1}{xe^{x^2}} \, dx = \int\limits^1_0 {\frac{e^{-x^2}}{x} \, dx
  2. [Integral] Rewrite [Improper Integral]:                                                         \displaystyle \int\limits^1_0 {\frac{1}{xe^{x^2}} \, dx = \lim_{a \to 0^+} \int\limits^1_a {\frac{e^{-x^2}}{x} \, dx

<u>Step 3: Integrate Pt. 2</u>

<em>Identify variables for u-substitution.</em>

  1. Set:                                                                                                                 \displaystyle u = -x^2
  2. Differentiate [Basic Power Rule]:                                                                 \displaystyle \frac{du}{dx} = -2x
  3. [Derivative] Rewrite:                                                                                     \displaystyle du = -2x \ dx

<em>Rewrite u-substitution to format u-solve.</em>

  1. Rewrite <em>du</em>:                                                                                                     \displaystyle dx = \frac{-1}{2x} \ dx

<u>Step 4: Integrate Pt. 3</u>

  1. [Integral] Rewrite [Integration Property - Multiplied Constant]:                 \displaystyle \int\limits^1_0 {\frac{1}{xe^{x^2}} \, dx = \lim_{a \to 0^+} -\int\limits^1_a {-\frac{e^{-x^2}}{x} \, dx
  2. [Integral] Substitute in variables:                                                                 \displaystyle \int\limits^1_0 {\frac{1}{xe^{x^2}} \, dx = \lim_{a \to 0^+} -\int\limits^1_a {\frac{e^{u}}{-2u} \, du
  3. [Integral] Rewrite [Integration Property - Multiplied Constant]:                 \displaystyle \int\limits^1_0 {\frac{1}{xe^{x^2}} \, dx = \lim_{a \to 0^+} \frac{1}{2}\int\limits^1_a {\frac{e^{u}}{u} \, du
  4. [Integral] Substitute [Exponential Integral Function]:                                 \displaystyle \int\limits^1_0 {\frac{1}{xe^{x^2}} \, dx = \lim_{a \to 0^+} \frac{1}{2}[Ei(u)] \bigg| \limits^1_a
  5. Back-Substitute:                                                                                             \displaystyle \int\limits^1_0 {\frac{1}{xe^{x^2}} \, dx = \lim_{a \to 0^+} \frac{1}{2}[Ei(-x^2)] \bigg| \limits^1_a
  6. Evaluate [Integration Rule - FTC 1]:                                                             \displaystyle \int\limits^1_0 {\frac{1}{xe^{x^2}} \, dx = \lim_{a \to 0^+} \frac{1}{2}[Ei(-1) - Ei(a)]
  7. Simplify:                                                                                                         \displaystyle \int\limits^1_0 {\frac{1}{xe^{x^2}} \, dx = \lim_{a \to 0^+} \frac{Ei(-1) - Ei(a)}{2}
  8. Evaluate limit [Limit Rule - Variable Direct Substitution]:                           \displaystyle \int\limits^1_0 {\frac{1}{xe^{x^2}} \, dx = \infty

∴  \displaystyle \int\limits^1_0 {\frac{1}{xe^{x^2}} \, dx  diverges.

Topic: Multivariable Calculus

7 0
3 years ago
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