Integral Calculus
Assumed Knowledge
Reminder — Standard Integrals
Note — Four Basic Rules
Standard Integrals of e x ,
1
x
and sec2 x
Definition — Standard Integrals of e x , x1 and sec2 x
Example — Standard Integrals of e x , x1 and sec2 x
5.1 Assumed Knowledge
Reminder — Standard Integrals
5.1 Assumed Knowledge
Reminder — Standard Integrals
I
R
ax n dx =
a
x n+1 + c.
n+1
5.1 Assumed Knowledge
Reminder — Standard Integrals
I
I
a
x n+1 + c.
n+1
R
1
cos(ax + b) dx = sin(ax + b) + c.
a
R
ax n dx =
5.1 Assumed Knowledge
Reminder — Standard Integrals
I
I
I
a
x n+1 + c.
n+1
R
1
cos(ax + b) dx = sin(ax + b) + c.
a
R
1
sin(ax + b) dx = − cos(ax + b) + c.
a
R
ax n dx =
5.1 Assumed Knowledge
Reminder — Standard Integrals
a
x n+1 + c.
n+1
R
1
I
cos(ax + b) dx = sin(ax + b) + c.
a
R
1
I
sin(ax + b) dx = − cos(ax + b) + c.
a
The notation F(x) is Rused to represent the antiderivative of f (x) so
that F0 (x) = f (x) or f (x) dx = F(x) + c.
I
R
ax n dx =
Note — Four Basic Rules
Note — Four Basic Rules
1.
R
R
R
R
(af (x)) + (bg (x)) dx = a f (x) dx + b g (x) dx.
Note — Four Basic Rules
1.
R
R
R
R
(af (x)) + (bg (x)) dx = a f (x) dx + b g (x) dx.
2.
Rb
a
f (x) dx = F(b) − F(a) where F0 (x) = f (x).
Note — Four Basic Rules
1.
R
R
R
R
(af (x)) + (bg (x)) dx = a f (x) dx + b g (x) dx.
2.
Rb
3.
Rc
a
a
f (x) dx = F(b) − F(a) where F0 (x) = f (x).
f (x) dx =
Rb
a
f (x) dx +
Rc
b
f (x) dx, where a < b < c.
Note — Four Basic Rules
1.
R
R
R
R
(af (x)) + (bg (x)) dx = a f (x) dx + b g (x) dx.
2.
Rb
3.
Rc
4.
Rb
a
a
a
f (x) dx = F(b) − F(a) where F0 (x) = f (x).
f (x) dx =
Rb
a
f (x) dx = −
f (x) dx +
Ra
b
f (x) dx.
Rc
b
f (x) dx, where a < b < c.
Note — Four Basic Rules
1.
R
R
R
R
(af (x)) + (bg (x)) dx = a f (x) dx + b g (x) dx.
2.
Rb
3.
Rc
4.
Rb
a
a
a
f (x) dx = F(b) − F(a) where F0 (x) = f (x).
f (x) dx =
Rb
a
f (x) dx = −
f (x) dx +
Ra
b
Rc
b
f (x) dx, where a < b < c.
f (x) dx.
Item 2 is often referred to as the Fundamental Theorem of
Calculus. The integral represents the area under the curve f (x)
between the points x = a and x = b.
5.2 Standard Integrals of e x ,
1
x
and sec2 x
Definition — Standard Integrals of e x ,
1
x
and sec2 x
5.2 Standard Integrals of e x ,
1
x
and sec2 x
Definition — Standard Integrals of e x ,
I
R
e x dx = e x + c.
1
x
and sec2 x
5.2 Standard Integrals of e x ,
1
x
and sec2 x
Definition — Standard Integrals of e x ,
I
R
e x dx = e x + c.
I
R
1
x
dx = ln |x| + c.
1
x
and sec2 x
5.2 Standard Integrals of e x ,
1
x
and sec2 x
Definition — Standard Integrals of e x ,
I
R
e x dx = e x + c.
I
R
1
x
I
R
sec2 x dx = tan x + c.
dx = ln |x| + c.
1
x
and sec2 x
5.2 Standard Integrals of e x ,
1
x
and sec2 x
Definition — Standard Integrals of e x ,
I
R
e x dx = e x + c.
I
R
1
x
I
R
sec2 x dx = tan x + c.
1
x
and sec2 x
dx = ln |x| + c.
Note |x| is the modulus of x and |x| = x, when x ≥ 0 but
|x| = −x, when x ≤ 0.
Example — Standard Integrals of e x ,
1
x
and sec2 x
Example — Standard Integrals of e x ,
1
x
and sec2 x
Find the following indefinite and definite integrals.
R
1. (e 2x + 5x) dx
Example — Standard Integrals of e x ,
1
x
and sec2 x
Find the following indefinite and definite integrals.
R
1. (e 2x + 5x) dx
R
R
= e 2x dx + 5x dx
Example — Standard Integrals of e x ,
1
x
and sec2 x
Find the following indefinite and definite integrals.
R
1. (e 2x + 5x) dx
R
R
= e 2x dx + 5x dx
= 12 e 2x + 25 x 2 + c
Example — Standard Integrals of e x ,
1
x
and sec2 x
Find the following indefinite and definite integrals.
R
1. (e 2x + 5x) dx
R
R
= e 2x dx + 5x dx
= 12 e 2x + 25 x 2 + c
2.
R
1
dx
5x + 4
Example — Standard Integrals of e x ,
1
x
and sec2 x
Find the following indefinite and definite integrals.
R
1. (e 2x + 5x) dx
R
R
= e 2x dx + 5x dx
= 12 e 2x + 25 x 2 + c
2.
R
1
dx
5x + 4
=
1
5
ln |5x + 4| + c
3.
R
π
4
0
1 + sin2 x
cos2 x
dx
1 + sin2 x
dx
3. 0
cos2 x
Rπ
R π4 sin2 x
1
= 04
dx
+
dx
0
cos2 x
cos2 x
R
π
4
1 + sin2 x
dx
3. 0
cos2 x
Rπ
R π4 sin2 x
1
= 04
dx
+
dx
0
cos2 x
cos2 x
R
π
4
=
R
π
4
0
sec2 x dx +
R
π
4
0
tan2 x dx
1 + sin2 x
dx
3. 0
cos2 x
Rπ
R π4 sin2 x
1
= 04
dx
+
dx
0
cos2 x
cos2 x
R
π
4
=
R
π
4
=
R
π
4
0
0
sec2 x dx +
R
π
4
sec2 x dx +
R
π
4
0
0
tan2 x dx
sec2 x − 1 dx
1 + sin2 x
dx
3. 0
cos2 x
Rπ
R π4 sin2 x
1
= 04
dx
+
dx
0
cos2 x
cos2 x
R
π
4
=
R
π
4
=
R
π
4
0
0
=2
R
sec2 x dx +
R
π
4
sec2 x dx +
R
π
4
sec2 x dx −
R
π
4
0
0
0
tan2 x dx
sec2 x − 1 dx
π
4
0
1 dx
1 + sin2 x
dx
3. 0
cos2 x
Rπ
R π4 sin2 x
1
= 04
dx
+
dx
0
cos2 x
cos2 x
R
π
4
=
R
π
4
=
R
π
4
0
0
=2
R
sec2 x dx +
R
π
4
sec2 x dx +
R
π
4
sec2 x dx −
R
π
4
0
π
0
π
= 2[tan x]04 − [x]04
0
tan2 x dx
sec2 x − 1 dx
π
4
0
1 dx
1 + sin2 x
dx
3. 0
cos2 x
Rπ
R π4 sin2 x
1
= 04
dx
+
dx
0
cos2 x
cos2 x
R
π
4
=
R
π
4
=
R
π
4
0
0
=2
R
sec2 x dx +
R
π
4
sec2 x dx +
R
π
4
sec2 x dx −
R
π
4
0
π
0
0
tan2 x dx
sec2 x − 1 dx
π
4
0
1 dx
π
= 2[tan x]04 − [x]04
= 2(tan π4 − tan 0) − ( π4 − 0)
1 + sin2 x
dx
3. 0
cos2 x
Rπ
R π4 sin2 x
1
= 04
dx
+
dx
0
cos2 x
cos2 x
R
π
4
=
R
π
4
=
R
π
4
0
0
=2
R
sec2 x dx +
R
π
4
sec2 x dx +
R
π
4
sec2 x dx −
R
π
4
0
π
0
0
tan2 x dx
sec2 x − 1 dx
π
4
0
1 dx
π
= 2[tan x]04 − [x]04
= 2(tan π4 − tan 0) − ( π4 − 0)
=2−
π
4
Further Examples
Maths In Action: Book 1
Page 72
Exercise 2A/2B