Question

True or False.

If

F(x) = ∫ -23x sin(t) dt

then the second fundamental theorem of calculus can be used to evaluate F '(x) as follows

F '(x) = sin (3x)

Answer 1

False.

General Formulas and Concepts:

Calculus

Differentiation

• Derivatives
• Derivative Notation

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

Basic Power Rule:

1. f(x) = cxⁿ
2. f’(x) = c·nxⁿ⁻¹

Derivative Rule [Chain Rule]:
`\displaystyle (d)/(dx)[f(g(x))] =f'(g(x)) \cdot g'(x)`

Integration

• Integrals
• Definite Integrals
• Integration Constant C

Integration Rule [Fundamental Theorem of Calculus 2]:
`\displaystyle (d)/(dx)[\int\limits^x_a {f(t)} \, dt] = f(x)`

Explanation:

Step 1: Define

Identify

`\displaystyle F(x) = \int\limits^(3x)_(-2) {sin(t)} \, dt`

Step 2: Differentiate

1. Chain Rule:
   `\displaystyle F'(x) = (d)/(dx)[\int\limits^(3x)_(-2) {sin(t)} \, dt] \cdot (d)/(dx)[3x]`
2. Rewrite [Derivative Property - Multiplied Constant]:
   `\displaystyle F'(x) = (d)/(dx)[\int\limits^(3x)_(-2) {sin(t)} \, dt] \cdot 3(d)/(dx)[x]`
3. Basic Power Rule:
   `\displaystyle F'(x) = (d)/(dx)[\int\limits^(3x)_(-2) {sin(t)} \, dt] \cdot 3x^(1 - 1)`
4. Simplify:
   `\displaystyle F'(x) = 3(d)/(dx)[\int\limits^(3x)_(-2) {sin(t)} \, dt]`
5. Integration Rule [Fundamental Theorem of Calculus 2]:
   `\displaystyle F'(x) = 3sin(3x)`

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

Unit: Integration

Book: College Calculus 10e