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