MATH 155/GRACEY
INTEGRAL REVIEW
THEOREM: THE CONSTANT RULE
Let k be a real number.
∫ kdx = x + C
Example 1: Find the indefinite integral.
∫ −3dx
THEOREM: THE POWER RULE
Let n be a rational number.
x n +1
∫ x dx = n + 1 + C , n ≠ −1
n
Example 2: Find the following indefinite integrals.
a.
∫x
−5
dx
b.
∫x
12
dx
c.
∫x
−2 3
dx
THEOREM: THE CONSTANT MULTIPLE RULE
If f is an integrable function and c is a real number, then cf is also
integrable and
∫ cf ( x ) dx = c ∫ f ( x )dx
MATH 155/GRACEY
INTEGRAL REVIEW
3
Example 3: Find the area of the region bounded by f ( x ) = 2 x , x = 1 , x = 3 ,
and y = 0 .
THEOREM: THE SUM AND DIFFERENCE RULES
The sum (or difference) of two integrable functions f and g is itself
integrable. Moreover, the antiderivative of f + g (or f − g ) is the sum (or
difference) of the antiderivatives of f and g .
∫  f ( x ) + g ( x )dx = ∫ f ( x ) dx + ∫ g ( x ) dx
∫  f ( x ) − g ( x ) dx = ∫ f ( x ) dx − ∫ g ( x ) dx
Example 4: Find the indefinite integral.
a.
 x − 5x2 
∫  x dx


b.
∫(x
3
2
+ 1) dx
MATH 155/GRACEY
INTEGRAL REVIEW
THEOREM: ANTIDERIVATIVES OF THE TRIGONOMETRIC
FUNCTIONS
∫ sin xdx = − cos x + C
∫ csc x cot xdx = − csc x + C
∫ sec xdx = tan x + C
∫ tan xdx = − ln cos x + C
∫ sec xdx = ln sec x + tan x + C
2
∫ cos xdx = sin x + C
∫ sec x tan xdx = sec x + C
∫ csc xdx = − cot x + C
∫ cot xdx = ln sin x + C
∫ csc xdx = − ln csc x + cot x + C
2
Example 5: Integrate.
a.
b.
∫ sin
2
xdx
∫ ( − cscθ + cscθ cot θ ) dθ
c.
∫ 3tan xdx
1
dθ
d. ∫ 1 + cos θ
MATH 155/GRACEY
INTEGRAL REVIEW
THEOREM: ANTIDIFFERENTIATION OF A COMPOSITE FUNCTION
Let g be a function whose range is an interval I and let f be a function
that is continuous on I . If g is differentiable on its domain and F is an
antiderivative of f on I , then
∫ f ( g ( x ) ) g ′ ( x ) dx = F ( g ( x ) ) + C
Letting u = g ( x ) gives du = g ′ ( x ) dx and
∫ f ( u ) du = F ( u ) + C
Example 6: Find the following definite and indefinite integrals.
a.
∫(x
)
b.
2
∫ x ( 5 − 2 x ) dx
1 − x dx
5
MATH 155/GRACEY
INTEGRAL REVIEW
c.
∫ cos
d.
 4 + 5x 3 2
∫  x

5
e.
2
3xdx

 dx


5 + 6x + x2
∫3 5 + x dx
MATH 155/GRACEY
INTEGRAL REVIEW
2
f.
∫
x − 1 dx
0
π
g.
3
∫π tan
3
x sec 2 xdx
4
Theorem: LOG RULE FOR INTEGRATION
Let u be a differentiable function of x.
1.
1
∫ x dx = ln x + C
2.
1
∫ u du = ln u + C
MATH 155/GRACEY
INTEGRAL REVIEW
Theorem: INTEGRATION RULES FOR EXPONENTIAL FUNCTIONS
Let u be a differentiable function of x.
1.
∫ e dx = e
x
x
+C
2.
∫ e du = e
u
u
+C
 1  x
3. ∫ a x dx = 
 a + C , a is a positive real number, a ≠ 1
ln
a


Example 7: Find the following definite and indefinite integrals.
a.
b.
5t 2 − t − 1
∫ 2 − t dx
∫
5
(
x ln x
)
2
2
c.
dx
d.
1

x
+
∫  x  dx
∫
1
x
2
3
(1 + x )
1
3
dx
MATH 155/GRACEY
e.
f.
e x + e− x
dx
e x − e− x
∫
2
∫
2e
1
0
π 2
g.
x
dx
1− x
∫π ( sec x )dx
3
2
INTEGRAL REVIEW
h.
∫
π
1
1


3
−
+ tan 2 x dx

2x


π 2
i.
∫π
j.
−x
2
∫ dx
−
2
sin x cos 2 xdx