Question

How do I evaluate this definite integral?

Answer 1

`\displaystyle \int\limits^e_1 {(8)/(x)} \, dx = 8`

General Formulas and Concepts:

Calculus

Integration

• Integrals

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^e_1 {(8)/(x)} \, dx`

Step 2: Integrate

1. [Integral] Rewrite [Integration Property - Multiplied Constant]:
   `\displaystyle \int\limits^e_1 {(8)/(x)} \, dx = 8\int\limits^e_1 {(1)/(x)} \, dx`
2. [Integral] Logarithmic Integration:
   `\displaystyle \int\limits^e_1 {(8)/(x)} \, dx = 8 \ln |x| \bigg| \limits^e_1`
3. Evaluate [Integration Rule - Fundamental Theorem of Calculus 1]:
   `\displaystyle \int\limits^e_1 {(8)/(x)} \, dx = 8`

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

Unit: Integration