Spm Addmath_integration Exercise
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ONE-SCHOOL.NET
Integration 3.1 Integration as the Inverse of Differentiation, Integration of axn and integration of the Functions of the Sum/Difference of Algebraic Terms
2. Find a. ∫ x 4 dx =
b.
∫ xdx =
c.
∫x
d.
∫ x 2 dx =
e.
∫ x 5 dx =
∫ adx = ax + C Example ∫ 2dx = 2 x + C
Exercise 3.1:
−2
dx =
1
1. Find a. ∫ 4dx
4
b.
∫120dx
c.
∫ −12dx
d.
Example 1 x −4 −5 dx = x dx = +C ∫ x5 ∫ −4
2
∫ 3 dx
∫x
g.
∫x
1 e. ∫ − dx 4
x n+1 ∫ x dx = n + 1 + C n
Example x3 1. ∫ x dx = + C 3 2
1
1
f.
3
1 7
dx =
dx =
ONE-SCHOOL.NET Example 3
c.
∫ −130 x
d.
∫ 4 x dx =
e.
∫ − 15 x dx =
f.
5x4 ∫ 9 dx =
g.
∫−
550
dx =
3
1 2
x2 2x 2 xdx = ∫ x dx = +C = +C 3 3 2
∫
∫
h.
∫
i.
xdx =
3
2
x 3 dx =
7
6
ax n+1 ∫ ax dx = n + 1 + C n
Example 2 x4 x4 ∫ 2 x dx = 4 + C = 2 + C 3
3. Find a. ∫ 3x 5dx =
b.
∫ −7 x dx = 3
2
2 x5 dx = 7
ONE-SCHOOL.NET x −3 dx = h. ∫ 2
i.
1
3x −2 ∫ 5 dx =
l.
5 ∫ − 7 x 3 dx =
m.
4 − 34 ∫ 7 x dx =
1
j.
4x 2 ∫ 11 dx =
Example 2 2 −5 2 x −4 ∫ 3x5 dx = ∫ 3 x dx = 3 ( −4 ) + C 2 x −4 x −4 )+C = +C = ( 3 −4 −6 2
n.
∫x
o.
∫x
2
dx =
2
2x 3 k. ∫ dx = 3
3
7 3
dx =
ONE-SCHOOL.NET p.
−2
∫x
3
dx =
Example 1
2 xdx = ∫ 2 x 2 dx
∫
3
3
2 2x 2 x2 = 2( ) + C = +C 3 3 2
q.
r.
s.
2
∫ 5x
5
3
5
6
∫2
u.
∫ −7
v.
∫3
w.
∫
xdx =
dx =
∫ − 4x
∫ 7x
t.
7
x 3 dx =
dx =
dx =
4
2
xdx =
3xdx =
ONE-SCHOOL.NET x.
∫
4x 5 dx =
∫ (u ± v)dx = ∫ udx ± ∫ vdx u and v are functions in x
Example 1. ∫ 3x 2 + 2 xdx = ∫ 3x 2 dx + ∫ 2 xdx
3x3 2 x 2 3 x3 2 x 2 = + +C = + +C 3 2 3 2 = x3 + x 2 + C
y.
z.
∫
∫
3
4
4. Find a. ∫ x3 + 2dx
6x 2 dx =
b.
∫x
c.
∫ 2x
d.
∫12 x
e.
∫2x
2xdx =
5
2
1
+ 2 xdx
− 5 x 3 + 1dx
2
7
− 8 x −3dx
3
+ 9 x8dx
ONE-SCHOOL.NET f.
1
∫x
3
+
2 dx x4
Example 2 ∫ ( x + 2)(3x + 1)dx = ∫ 3x + 7 x + 2dx = ∫ 3 x 2 dx + ∫ 7 xdx + ∫ 2dx 3x3 7 x 2 + + 2x + C 3 2 7 x2 = x3 + + 2x + C 2 =
g.
h.
1
∫ 2x
3
+
i.
∫ (2 x + 3) dx
j.
∫ x (5x + 3)dx
k.
∫ (3x + 1)( x
2
9 8 x dx 2
2
1 2 + x 9 ∫ 3x5 2 + 3dx
6
2
− 2)dx
ONE-SCHOOL.NET l.
∫ x( x
3
3 + )dx x
Example 3x3 + x 2 − x dx = ∫ 3 x 2 + x − 1dx ∫ x = ∫ 3 x 2 dx + ∫ xdx − ∫ 1dx 3x3 x 2 = + − x+C 3 2 2 x = x3 + − x + C 2
m.
n.
∫ (x
2
o.
x 2 − 3x ∫ x dx
p.
x5 − 3x3 + 5 ∫ x 2 dx
+ 5) 2 dx
∫ ( x + 1)( x
2
+ 3x − 2)dx
(2 x − 1) 2 dx q. ∫ x
7
ONE-SCHOOL.NET r.
3.2 Integration by Substitution
(2 x 3 − 2)( x + 5) dx ∫ x2
(ax + b) n+1 ∫ (ax + b) dx = a(n + 1) + C Example (3 x + 5) 4 3 ∫ (3x + 5) dx = 3(4) + C n
=
(3 x + 5) 4 +C 12
Exercise 3.2: 1. Find a. ∫ ( x − 1)5 dx
s.
( x 2 − 3x)( x + 5) dx ∫ x3
8
b.
∫ (5x − 9) dx
c.
∫ (12 x + 3) dx
3
7
ONE-SCHOOL.NET d.
∫ (2 x − 18)
e.
∫ (7 x + 6) 2 dx
−2
dx
i.
∫
2 x + 5dx
j.
∫
9 x − 4dx
5
1
f.
∫ (2 x − 3)
g.
∫ (5x + 7)
h.
∫ 3(9 x − 2)
2
dx
3
dx
2
2
5
dx
9
Integration 3.1 Integration as the Inverse of Differentiation, Integration of axn and integration of the Functions of the Sum/Difference of Algebraic Terms
2. Find a. ∫ x 4 dx =
b.
∫ xdx =
c.
∫x
d.
∫ x 2 dx =
e.
∫ x 5 dx =
∫ adx = ax + C Example ∫ 2dx = 2 x + C
Exercise 3.1:
−2
dx =
1
1. Find a. ∫ 4dx
4
b.
∫120dx
c.
∫ −12dx
d.
Example 1 x −4 −5 dx = x dx = +C ∫ x5 ∫ −4
2
∫ 3 dx
∫x
g.
∫x
1 e. ∫ − dx 4
x n+1 ∫ x dx = n + 1 + C n
Example x3 1. ∫ x dx = + C 3 2
1
1
f.
3
1 7
dx =
dx =
ONE-SCHOOL.NET Example 3
c.
∫ −130 x
d.
∫ 4 x dx =
e.
∫ − 15 x dx =
f.
5x4 ∫ 9 dx =
g.
∫−
550
dx =
3
1 2
x2 2x 2 xdx = ∫ x dx = +C = +C 3 3 2
∫
∫
h.
∫
i.
xdx =
3
2
x 3 dx =
7
6
ax n+1 ∫ ax dx = n + 1 + C n
Example 2 x4 x4 ∫ 2 x dx = 4 + C = 2 + C 3
3. Find a. ∫ 3x 5dx =
b.
∫ −7 x dx = 3
2
2 x5 dx = 7
ONE-SCHOOL.NET x −3 dx = h. ∫ 2
i.
1
3x −2 ∫ 5 dx =
l.
5 ∫ − 7 x 3 dx =
m.
4 − 34 ∫ 7 x dx =
1
j.
4x 2 ∫ 11 dx =
Example 2 2 −5 2 x −4 ∫ 3x5 dx = ∫ 3 x dx = 3 ( −4 ) + C 2 x −4 x −4 )+C = +C = ( 3 −4 −6 2
n.
∫x
o.
∫x
2
dx =
2
2x 3 k. ∫ dx = 3
3
7 3
dx =
ONE-SCHOOL.NET p.
−2
∫x
3
dx =
Example 1
2 xdx = ∫ 2 x 2 dx
∫
3
3
2 2x 2 x2 = 2( ) + C = +C 3 3 2
q.
r.
s.
2
∫ 5x
5
3
5
6
∫2
u.
∫ −7
v.
∫3
w.
∫
xdx =
dx =
∫ − 4x
∫ 7x
t.
7
x 3 dx =
dx =
dx =
4
2
xdx =
3xdx =
ONE-SCHOOL.NET x.
∫
4x 5 dx =
∫ (u ± v)dx = ∫ udx ± ∫ vdx u and v are functions in x
Example 1. ∫ 3x 2 + 2 xdx = ∫ 3x 2 dx + ∫ 2 xdx
3x3 2 x 2 3 x3 2 x 2 = + +C = + +C 3 2 3 2 = x3 + x 2 + C
y.
z.
∫
∫
3
4
4. Find a. ∫ x3 + 2dx
6x 2 dx =
b.
∫x
c.
∫ 2x
d.
∫12 x
e.
∫2x
2xdx =
5
2
1
+ 2 xdx
− 5 x 3 + 1dx
2
7
− 8 x −3dx
3
+ 9 x8dx
ONE-SCHOOL.NET f.
1
∫x
3
+
2 dx x4
Example 2 ∫ ( x + 2)(3x + 1)dx = ∫ 3x + 7 x + 2dx = ∫ 3 x 2 dx + ∫ 7 xdx + ∫ 2dx 3x3 7 x 2 + + 2x + C 3 2 7 x2 = x3 + + 2x + C 2 =
g.
h.
1
∫ 2x
3
+
i.
∫ (2 x + 3) dx
j.
∫ x (5x + 3)dx
k.
∫ (3x + 1)( x
2
9 8 x dx 2
2
1 2 + x 9 ∫ 3x5 2 + 3dx
6
2
− 2)dx
ONE-SCHOOL.NET l.
∫ x( x
3
3 + )dx x
Example 3x3 + x 2 − x dx = ∫ 3 x 2 + x − 1dx ∫ x = ∫ 3 x 2 dx + ∫ xdx − ∫ 1dx 3x3 x 2 = + − x+C 3 2 2 x = x3 + − x + C 2
m.
n.
∫ (x
2
o.
x 2 − 3x ∫ x dx
p.
x5 − 3x3 + 5 ∫ x 2 dx
+ 5) 2 dx
∫ ( x + 1)( x
2
+ 3x − 2)dx
(2 x − 1) 2 dx q. ∫ x
7
ONE-SCHOOL.NET r.
3.2 Integration by Substitution
(2 x 3 − 2)( x + 5) dx ∫ x2
(ax + b) n+1 ∫ (ax + b) dx = a(n + 1) + C Example (3 x + 5) 4 3 ∫ (3x + 5) dx = 3(4) + C n
=
(3 x + 5) 4 +C 12
Exercise 3.2: 1. Find a. ∫ ( x − 1)5 dx
s.
( x 2 − 3x)( x + 5) dx ∫ x3
8
b.
∫ (5x − 9) dx
c.
∫ (12 x + 3) dx
3
7
ONE-SCHOOL.NET d.
∫ (2 x − 18)
e.
∫ (7 x + 6) 2 dx
−2
dx
i.
∫
2 x + 5dx
j.
∫
9 x − 4dx
5
1
f.
∫ (2 x − 3)
g.
∫ (5x + 7)
h.
∫ 3(9 x − 2)
2
dx
3
dx
2
2
5
dx
9
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