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
FunctionsFunction Notation
Calculus
Derivatives
Derivative Notation
Antiderivatives - Integrals
Integration Constant C
Integration Rule [Reverse Power Rule]:
Integration Property [Addition/Subtraction]:
Integration Property [Multiplied Constant]:
Step-by-step explanation:
Step 1: Define
Identify
Step 2: Integration
Integrate the derivative to find function.
[Derivative] Integrate:
Simplify:
Rewrite [Integration Property - Addition/Subtraction]:
[1st Integral] Integrate [Integral Rule - Reverse Power Rule]:
[2nd Integral] Integrate [Integral Rule - Reverse Power Rule]:
[3rd Integral] Rewrite [Integral Property - Multiplied Constant]:
[3rd Integral] Integrate:
Our general solution is .
Step 3: Find Particular Solution
Find Integration Constant C for function using given condition.
Substitute in condition [Function]:
Substitute in function value:
Evaluate exponents:
Evaluate natural log:
Multiply:
Add:
Simplify:
[Subtraction Property of Equality] Isolate C:
Rewrite:
Substitute in C [Function]:
∴ Our particular solution to the differential equation is .
Topic: AP Calculus AB/BC (Calculus I/II)
Unit: Integration
Book: College Calculus 10e