Antiderivatives and Indefinite Integrals A Integration Example 1. Find a function f (x) such that slope of the Differentiation is the process of finding f ' ( x) given tangent line at point P( x, f ( x)) is equal to x . f (x) . Integration is the inverse process of differentiation. It means finding f (x) given f ' ( x) . f ( x) → differentiation → ← integration ← Example 2. Find a function f (x) such that f ' ( x) = mT ( x) = IRC ( x) instantaneous rate of change at x is 1 / x . Notes. f ' ( x) s (t ) → differentiation → f ' ' ( x) ← integration ← Example 3. Find a position function s (t ) such that the acceleration function is a (t ) = t . → differentiation → → differentiation → v (t ) a(t ) ← integration ← ← integration ← B Antiderivative Example 4. For each function given below, find an antiderivative function. Antiderivative of the function f (x) is a function or a) f ( x) = sin x F (x) such that F ' ( x) = f ( x) . b) f ( x) = 1 x c) f ( x) = e x C Families of Antiderivatives Example 4. For each function given below, find the corresponding family of antiderivative function.s If F (x) is an antiderivative of f (x) , so is also F ( x) + C where C is a constant called the constant of integration. a) f ( x) = cos x b) f ( x ) = x c) f ( x) = 1 1 − x2 D Initial Condition Example 5. From a height of 100m , a rock is launched vertically upward with a speed of 20m / s . Earth gravity An antiderivative of a function may be uniquely identified by an initial condition: accelerates downward the rock at 10m / s 2 . Find the velocity function and the position function. F (a) = b Designed by Iulia & Teodoru Gugoiu - Page 1 of 2 E Indefinite Integrals Example 6. Prove (by differentiation) that the following formulas are correct. Indefinite Integral is the most used notation for antiderivatives. If F (x) is an antiderivative of f (x) then we write: F ( x) = ∫ x(1 + 2 ln x)dx = x b) ∫ sin c) ∫ xe dx = ( x − 1)e 2 ln x + C ∫ f ( x)dx By definition: d dx a) 2 xdx = 2 x − sin 2 x +C 4 ∫ f ( x)dx = f ( x) Note. A better understanding of this notation is coming soon. x x +C G List of Indefinite Integrals ∫ dx = x + C ∫ sin xdx = − cos x + C ∫ cos xdx = sin x + C ∫ sec xdx = tan x + C ∫ csc xdx = − cot x + C x n +1 + C ; n ≠ −1 n +1 ∫ 1 ∫ x dx = ln | x | +C ∫ e dx = e + C x n dx = x ∫ x x a x dx = a +C ln a 2 ∫ sec x tan xdx = sec x + C ∫ csc x cot xdx = − csc x + C 1 ∫ x + 1 dx = tan x + C 2 ∫ F Properties of Antiderivatives or Indefinite Integrals −1 2 1− x 2 dx = sin −1 x + C Example 7. For each case, find the general indefinite integral. ∫ cf ( x)dx = c∫ f ( x)dx a) ∫ (x ∫ [ f ( x) ± g ( x)]dx = ∫ f ( x)dx ± ∫ g ( x)dx b) ∫ x −1 c) ∫ e x − e− x dx 2 d) ∫ (2 sin x − 3 cos x)dx Example 8. Find a function f (x) such that 1 2 x − x + 1)dx dx Example 9. Find f (x) given that f ' ' ' ( x) = x − 1 . f ' ' ( x) = −2 x + 1 , f ' (0) = 3 , and f (0) = 0 . Reading: (Stewart 6E) Pages 340-344 Homework: (Stewart 6E) Page 345 #1-63 (odd ) Page 397 #1-19 (odd) Designed by Iulia & Teodoru Gugoiu - Page 2 of 2