4 Integrals (JGGD)x - Integrals AERO BOARD PREP CALCULUS JGG DECENA Antidifferentiation AERO BOARD PREP CALCULUS JGG DECENA Illustration \u2022If F(x =

# 4 Integrals (JGGD)x - Integrals AERO BOARD PREP...

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AERO BOARD PREP 2014 CALCULUS JGG DECENA Integrals
AERO BOARD PREP 2014 CALCULUS JGG DECENA Antidifferentiation
AERO BOARD PREP 2014 CALCULUS JGG DECENA If F(x) = 10x, then F’(x) = 10. F(x) is the antiderivative of f(x) = 10 If F(x) = x 2 , F’(x) = 2x. F(x) is the antiderivative of f(x) = 2x Illustration
AERO BOARD PREP 2014 CALCULUS JGG DECENA Antidifferentiation
AERO BOARD PREP 2014 CALCULUS JGG DECENA Illustration F(x) = x 2 is an antiderivative of f(x) = 2x because F’(x)=f(x) But so are G(x) = x 2 + 2 and H(x) = x 2 – 7. Note that F, G and H differ only by a constant. For any constant real number, C, the function F(x)=x 2 + C is an antiderivative of f(x).
AERO BOARD PREP 2014 CALCULUS JGG DECENA Since the functions: G(x) = x 2 F(x) = x 2 + 2 H(x) = x 2 – 7 differ only by a constant, the slope of the tangent line remains the same The family of antiderivatives can be represented by F(x) + C Family of Antiderivatives
AERO BOARD PREP 2014 CALCULUS JGG DECENA The family of all antiderivaties of f(x) is indicated by Integral sign Integrand This is called the indefinite integral. The Indefinite Integral It is the most general antiderivative of f(x). f(x)dx
AERO BOARD PREP 2014 CALCULUS JGG DECENA The Indefinite Integral
AERO BOARD PREP 2014 CALCULUS JGG DECENA Sample Since 4 5 5x dx x C 5 4 d x 5x dx
AERO BOARD PREP 2014 CALCULUS JGG DECENA With the introduction of the new symbols, antidifferentiation becomes the same as getting the indefinite integral, that is, INTEGRATION .
AERO BOARD PREP 2014 CALCULUS JGG DECENA If we write a gets treated as a constant and x as the variable. If we write x gets treated as the constant. 2 2ax dx a(2x)dx ax C 2 2 2ax da a x C xa C
AERO BOARD PREP 2014 CALCULUS JGG DECENA Formula for Integration
AERO BOARD PREP 2014 CALCULUS JGG DECENA
AERO BOARD PREP 2014 CALCULUS JGG DECENA 2 2 sin cos cos sin sec tan csc cot sec tan sec csc cot csc xdx x C xdx x C dx x C dx x C x xdx x C x x dx x C   
AERO BOARD PREP 2014 CALCULUS JGG DECENA 1 1 2 2 2 2 sinh cosh cosh sinh tan ln | sec | cot ln | sin | 1 1 1 tan sin x dx x C x dx x C x dx x C xdx x C x x dx C dx C x a a a a a x
AERO BOARD PREP 2014 CALCULUS JGG DECENA
AERO BOARD PREP 2014 CALCULUS JGG DECENA Rules in Integration
AERO BOARD PREP 2014 CALCULUS JGG DECENA
AERO BOARD PREP 2014 CALCULUS JGG DECENA Example 2 3 4 5 z z dz 3 2 2 5 z z z C