Question: Differential Equations
Answer: A differential equation is one in which the unknown is a derivative
(dy/dx) = cos(x)
solution: y = sin(x) + C
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Question: How to find general solutions for differential equation
Answer: (1) General solution: find antiderivative of equations with C
(2) Explicit equation with initial condition: find antiderivative and plug into conditions to find C
(3) Solve using FTC
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Question: Information on slope fields
Answer: - Represent solutions to differential equations
- Because differential equations represent the slope of the tangent line to solution curve, we can calculate (dy/dx) at specific points and draw tangent segments at points
consistent veritically (same slope) = a function of x only
consistent horizontally (same slope) = a function of y only
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Question: Indefinite Integrals
Answer: - The function F is the antiderivative of f if F'(x) = f(x)
- F(x) + C is the antiderivative of f where C is the constant of integration
Formulas for indefinite integrals can be otained from differentiation
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Question: Rules of Integration
Answer: (1) (integral) x^n dx = (x^n+1 / n+1) + C, n/= -1
(2) (integral) k x f(x)dx = k (integral) f(x)dx
(3) (integral) [f(x) +/- g(x)]dx = (integral) f(x)dx +/- (integral)g(x)dx
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Question: Specific Rules of Integration for Particular Functions
Answer: (1) (integral) (1/u)du = ln(abs(u)) + C
(2) (integral) e^(u)du = e^u + C
(3) (integral) a^(u)du = (a^u/(ln(a))) + C
(4) (integral)sin(u)du = -cos(u) +C
(integral)cos(u)du = sin(u) +C
(integral)sec^2(u)du = tan(u) +C
(integral)csc^2(u)du = -cot(u) +C
(integral)sec(u)tan(u)du = sec(u) +C
(integral)csc(u)cot(u)du = -csc(u) +C
(5) (integral)(dx / (sqrt(1-(u^2)))du = sin^-1(u) +C
(integral)(dx / (1+(u^2))du = tan^-1(u) +C
(integral)(dx / (abs(x))(sqrt((u^2)-1))du = sec^-1(u) +C *if negative is cos^-1(u), cot^-1(u), or csc^-1(u) respectively
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Question: Antidifferentiation by Substitution
Answer: - Provides a means of integrating composite functions

let u = function being raised, find du/dx and solve to equal du
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Question: General Steps of Integration by Substitution
Answer: (1) let u = function being raised
(2) find du/dx and solve to equal du so it cancels other components in equation
(3) plug u back into equation with du
(4) Find antiderivative and add C, plug functon of u back in for answer
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Question: IMPORTANT FUNCTION FOR INTEGRATION BY SUBSTITUTION
Answer: (inetgral) tan(x)dx = (integral) (sinx/cosx) dx
u = cosx
-du = sinx(dx)
(integral) (-1 /u)du
Final Answer = - ln(cos(x)) + C or ln(sec(x)) + C
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Question: General Steps of Integration by Substitution with Defnite Integrals
Answer: (1) let u = function being raised
(2) find du/dx and solve to equal du so it cancels other components in equation
(3) Convert a and b by plugging them into u
(4) plug u back into equation with du
(5) Use FTC and add C, plug functon of u back in for answer
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Question: Info on Integration by Substution of Fractions
Answer: u = function in denominator ALWAYS
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