Question

What is the anti derivative of f'(x)=19x+15?​

Answer 1

Part A:
`\displaystyle f(x) = (19)/(2)x^2 + 15x + C`

Part B:
`\displaystyle f(x) = (19)/(2)x^2 + 15x - 5`

General Formulas and Concepts:

Pre-Algebra

Order of Operations: BPEMDAS

1. Brackets
2. Parenthesis
3. Exponents
4. Multiplication
5. Division
6. Addition
7. Subtraction

• Left to Right

Equality Properties

• Multiplication Property of Equality
• Division Property of Equality
• Addition Property of Equality
• Subtraction Property of Equality

Algebra I

Functions

• Function Notation

Calculus

Differentiation

• Derivatives
• Derivative Notation

Differential Equations

Integration

• Integrals
• [Indefinite Integrals] Integration Constant C

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

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

Integration Property [Addition/Subtraction]:
`\displaystyle \int {[f(x) \pm g(x)]} \, dx = \int {f(x)} \, dx \pm \int {g(x)} \, dx`

Explanation:

Step 1: Define

Identify

`\displaystyle f'(x) = 19x + 15`

Step 2: Find Antiderivative

1. [Derivative] Integrate both sides:
   `\displaystyle \int {f'(x)} \, dx = \int {19x + 15} \, dx`
2. [Left Integral] Simplify:
   `\displaystyle f(x) = \int {19x + 15} \, dx`
3. [Integral] Rewrite [Integration Property - Addition/Subtraction]:
   `\displaystyle f(x) = \int {19x} \, dx + \int {15} \, dx`
4. [Integrals] Rewrite [Integration Property - Multiplied Constant]:
   `\displaystyle f(x) = 19 \int {x} \, dx + 15 \int {} \, dx`
5. [Integrals] Integration Rule [Reverse Power Rule]:
   `\displaystyle f(x) = 19 \bigg( (x^2)/(2) \bigg) + 15x + C`
6. Simplify:
   `\displaystyle f(x) = (19)/(2)x^2 + 15x + C`

Step 3: Find Particular Solution

1. Substitute in function value [Function f(x)]:
   `\displaystyle 87 = (19)/(2)(-4)^2 + 15(-4) + C`
2. Evaluate:
   `\displaystyle 87 = 92 + C`
3. Solve:
   `\displaystyle C = -5`
4. Substitute in C [General Solution]:
   `\displaystyle f(x) = (19)/(2)x^2 + 15x - 5`

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

Unit: Differential Equations