Question

Calculus Question, just the first line. Other stuff is just my work...

Answer 1

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Answer 2

`\displaystyle \int\limits^(\infty)_{(\pi)/(2)} {\frac{7}{x^{(1)/(2)}}} \, dx = \infty`

General Formulas and Concepts:

Algebra I

• Exponential Rule [Rewrite]:
  `\displaystyle b^(-m) = (1)/(b^m)`

Calculus

Limits

Limit Rule [Variable Direct Substitution]:
`\displaystyle \lim_(x \to c) x = c`

Limit Property [Multiplied Constant]:
`\displaystyle \lim_(x \to c) bf(x) = b \lim_(x \to c) f(x)`

Integrals

• Definite Integrals
• Improper Integrals

Integration Constant C

Integration Rule [Reverse Power Rule]:
`\displaystyle \int {x^n} \, dx = (x^(n + 1))/(n + 1) + 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`

Explanation:

Step 1: Define

Identify

`\displaystyle \int\limits^(\infty)_{(\pi)/(2)} {\frac{7}{x^{(1)/(2)}}} \, dx`

Step 2: Integrate

1. [Integrand] Rewrite [Exponential Rule - Rewrite]:
   `\displaystyle \int\limits^(\infty)_{(\pi)/(2)} {\frac{7}{x^{(1)/(2)}}} \, dx = \int\limits^(\infty)_{(\pi)/(2)} {7x^{(-1)/(2)}} \, dx`
2. [Integral] Rewrite [Integration Property - Multiplied Constant]:
   `\displaystyle \int\limits^(\infty)_{(\pi)/(2)} {\frac{7}{x^{(1)/(2)}}} \, dx = 7\int\limits^(\infty)_{(\pi)/(2)} {x^{(-1)/(2)}} \, dx`
3. [Integral] Rewrite [Improper Integral]:
   `\displaystyle \int\limits^(\infty)_{(\pi)/(2)} {\frac{7}{x^{(1)/(2)}}} \, dx = \lim_(b \to \infty) 7\int\limits^b_{(\pi)/(2)} {x^{(-1)/(2)}} \, dx`
4. [Integral] Reverse Power Rule:
   `\displaystyle \int\limits^(\infty)_{(\pi)/(2)} {\frac{7}{x^{(1)/(2)}}} \, dx = \lim_(b \to \infty) 7(2x^{(1)/(2)}) \bigg| \limits^b_{(\pi)/(2)}`
5. [Limit] Rewrite [Limit Property - Multiplied Constant]:
   `\displaystyle \int\limits^(\infty)_{(\pi)/(2)} {\frac{7}{x^{(1)/(2)}}} \, dx = 7\lim_(b \to \infty) (2x^{(1)/(2)}) \bigg| \limits^b_{(\pi)/(2)}`
6. Evaluate [Integration Rule - Fundamental Theorem of Calculus 1]:
   `\displaystyle \int\limits^(\infty)_{(\pi)/(2)} {\frac{7}{x^{(1)/(2)}}} \, dx = 7\lim_(b \to \infty) (2b^{(1)/(2)} - √(2\pi))`
7. Evaluate limit [Limit Rule - Variable Direct Substitution]:
   `\displaystyle \int\limits^(\infty)_{(\pi)/(2)} {\frac{7}{x^{(1)/(2)}}} \, dx = \infty`

Topic: AP Calculus AB/BC (Calculus I/I + II)

Unit: Integration

Book: College Calculus 10e