Mathematics, 25.11.2020 14:00 ZOE1000000

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

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Answer from: miyah5316

Part A: $\displaystyle f(x) = \frac{19}{2}x^2 + 15x + C$

Part B: $\displaystyle f(x) = \frac{19}{2}x^2 + 15x - 5$

General Formulas and Concepts:

Pre-Algebra

Order of Operations: BPEMDAS

BracketsParenthesisExponentsMultiplicationDivisionAdditionSubtractionLeft to Right

Equality Properties

Multiplication Property of EqualityDivision Property of EqualityAddition Property of EqualitySubtraction Property of Equality

Algebra I

Functions

Function Notation

Calculus

Differentiation

DerivativesDerivative Notation

Differential Equations

Integration

Integrals[Indefinite Integrals] Integration Constant C

Integration Rule [Reverse Power Rule]: $\displaystyle \int {x^n} \, dx = \frac{x^{n + 1}}{n + 1} + C$

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

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

Step-by-step explanation:

Step 1: Define

Identify

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

Step 2: Find Antiderivative

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

Step 3: Find Particular Solution

Substitute in function value [Function f(x)]: $\displaystyle 87 = \frac{19}{2}(-4)^2 + 15(-4) + C$ Evaluate: $\displaystyle 87 = 92 + C$ Solve: $\displaystyle C = -5$ Substitute in C [General Solution]: $\displaystyle f(x) = \frac{19}{2}x^2 + 15x - 5$

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

Unit: Differential Equations

Answer from: Quest

I’m sorry i can’t u

Answer from: Quest

6/9 i need more characters in my answer

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