Math 152 – Spring 2016
Section 6.4
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Section 6.4– The Fundamental Theorem of
Calculus
Theorem (Fundamental Theorem of Calculus, Part 1). If f is continuous on [a, b],
then the function g defined by
Z x
f (t) dt, a ≤ x ≤ b
g(x) =
a
is continuous on [a, b] and differentiable on (a, b), and g 0 (x) = f (x).
Example 1. Find the derivatives of the following functions.
Rx
(a) g(x) = a cos2 (t) dt
(b) f (x) =
R sin x
2
t2 + 1 dt
Theorem (Fundamental Theorem of Calculus, Part II). If f is continuous on [a, b],
then
Z b
f (x) dx = F (b) − F (a)
a
where F is any antiderivate of f , that is F 0 = f .
Math 152 – Spring 2016
Section 6.4
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Example 2. Solve the following definite integrals.
R2
(a) 0 (3x3 − 8x + 2) dx
(b)
R π/3
0
sec x tan x dx
(c)
√
Z
3
2
√
−
1
2
√
6
dt
1 − t2
R
Definition. The notation f (x) dx is called the indefinite integral and is defined as
equal to the antiderivative. Therefore,
Z
f (x) dx = F (x) + C
where F 0 (x) = f (x) and C is a constant.
Note. There are two types of integrals.
1. The indefinite integral equals an antiderivative (a function with the +C).
Z
f (x) dx = F (x) + C, where F 0 (x) = f (x)
2. The definite integral equals a number.
Z a
f (x) dx = F (b) − F (a)
a
Math 152 – Spring 2016
Section 6.4
Table of Indefinite Integrals - You MUST have these MEMORIZED!
R
R
cf (x) dx = c f (x) dx
R
R
R
[f (x) + g(x)] dx = f (x) dx + g(x) dx
R n
n+1
x dx = xn+1 + C (n 6= −1)
R x
e dx = ex + C
R
sin x dx = − cos x + C
R 2
sec x dx = tan x + C
R
sec x tan x dx = sec x + C
R 1
−1
x+C
x2 +1 dx = tan
Example 3. Find the general indefinite integrals.
R 2 √x
(a) 3x −4
dx
2x
(b)
R
sin x +
(c)
R
5
x
1
x2 +1
+ 7x dx
+ sec2 x dx
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R
1
x
R
ax dx =
R
cos x dx = sin x + C
R
csc2 x dx = − cot x + C
R
csc x cot x dx = − csc x + C
R
√ 1
1−x2
= ln |x| + C
ax
ln a
+C
dx = sin−1 x + C
Math 152 – Spring 2016
Section 6.4
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Example 4. Suppose a particle is traveling at a velocity of v(t) = 2t2 − 4t in m/s.
(a) Find the displacement (distance from where it started to where it stopped) during
the time period 1 ≤ x ≤ 3.
(b) Find the total distance traveled during this time period.
Example 5. Suppose a particle has acceleration function a(t) = 3t − 4 and initial
velocity v(0) = 7 m/s. Find the velocity function.