Universiti Pendidikan Sultan Idris

1. Beginning Calculus
- Integration By Parts -
Shahrizal Shamsuddin Norashiqin Mohd Idrus
Department of Mathematics,
FSMT - UPSI
(LECTURE SLIDES SERIES)
VillaRINO DoMath, FSMT-UPSI
Integration By Parts 1 / 20

2. Algebraic Procedures Integration By Parts Partial Fractions
Learning Outcomes
Apply techniques of integration:
Algebraic procedures
Integration by parts
Partial fractions
VillaRINO DoMath, FSMT-UPSI
Integration By Parts 2 / 20

3. Algebraic Procedures Integration By Parts Partial Fractions
Simplifying Substitution - Example
Z
2x 9
p
x2 9x + 1
dx
VillaRINO DoMath, FSMT-UPSI
Integration By Parts 3 / 20

4. Algebraic Procedures Integration By Parts Partial Fractions
Completing The Square - Example
Z
dx
p
8x x2
VillaRINO DoMath, FSMT-UPSI
Integration By Parts 4 / 20

5. Algebraic Procedures Integration By Parts Partial Fractions
Reducing Improper Fraction - Example
Z
3x2 7x
3x + 2
dx
VillaRINO DoMath, FSMT-UPSI
Integration By Parts 5 / 20

6. Algebraic Procedures Integration By Parts Partial Fractions
Separating Fractions
Z
3x + 2
p
1 x2
dx
VillaRINO DoMath, FSMT-UPSI
Integration By Parts 6 / 20

7. Algebraic Procedures Integration By Parts Partial Fractions
The Formula
d
dx
(uv) = v
du
dx
+ u
dv
dx
d (uv) = vdu + udv
udv = d (uv) vdu
Z
udv = uv
Z
vdu
VillaRINO DoMath, FSMT-UPSI
Integration By Parts 7 / 20

8. Algebraic Procedures Integration By Parts Partial Fractions
The Formula
Z
udv = uv
Z
vdu
R
udv is expressed in terms of
R
vdu
with a proper choice of u and v, the second integral
R
vdu may be
easier to evaluate.
VillaRINO DoMath, FSMT-UPSI
Integration By Parts 8 / 20

9. Algebraic Procedures Integration By Parts Partial Fractions
Using IBP
WHEN:
if substitution does not work
HOW:
match
Z
f (x) g (x) dx with
Z
udv
u = f (x) , dv = g (x) dx or u = g (x) , dv = f (x) dx
VillaRINO DoMath, FSMT-UPSI
Integration By Parts 9 / 20

10. Algebraic Procedures Integration By Parts Partial Fractions
Example
Z
x cos xdx
Choice 1: u = 1, dv = x cos xdx
Choice 2: u = x, dv = cos xdx
Choice 3: u = x cos x, dv = dx
Choice 4: u = cos x.dv = xdx
VillaRINO DoMath, FSMT-UPSI
Integration By Parts 10 / 20

11. Algebraic Procedures Integration By Parts Partial Fractions
Example - continue
Choice 2:
u = x ) du = dx
dv = cos xdx ) v = sin x
Z
udv = uv
Z
vdu
)
Z
x cos xdx = x sin x
Z
sin xdx
= x sin x + cos x + C
VillaRINO DoMath, FSMT-UPSI
Integration By Parts 11 / 20

12. Algebraic Procedures Integration By Parts Partial Fractions
Example
Z
ln xdx
VillaRINO DoMath, FSMT-UPSI
Integration By Parts 12 / 20

13. Algebraic Procedures Integration By Parts Partial Fractions
Example - Repeated Use
Z
x2
ex
dx
VillaRINO DoMath, FSMT-UPSI
Integration By Parts 13 / 20

14. Algebraic Procedures Integration By Parts Partial Fractions
Example - Repeated Use
Z
ex
cos xdx
VillaRINO DoMath, FSMT-UPSI
Integration By Parts 14 / 20

15. Algebraic Procedures Integration By Parts Partial Fractions
Partial Fractions
The method of rewriting rational functions as sum of simpler fractions is
called the method of partial fractions.
VillaRINO DoMath, FSMT-UPSI
Integration By Parts 15 / 20

16. Algebraic Procedures Integration By Parts Partial Fractions
Two Distinct Linear Factors - Example
5x 3
x2 2x 3
=
A
x + 1
+
B
x 3
VillaRINO DoMath, FSMT-UPSI
Integration By Parts 16 / 20

17. Algebraic Procedures Integration By Parts Partial Fractions
A Repeated Linear Factor - Example
6x + 7
(x + 2)2
=
A
x + 2
+
B
(x + 2)2
VillaRINO DoMath, FSMT-UPSI
Integration By Parts 17 / 20

18. Algebraic Procedures Integration By Parts Partial Fractions
An Improper Fraction - Example
2x3 4x2 x 3
x2 2x 3
VillaRINO DoMath, FSMT-UPSI
Integration By Parts 18 / 20

19. Algebraic Procedures Integration By Parts Partial Fractions
An Irreducible Quadratic Factor - Example
2x + 4
(x2 + 1) (x 1)2
=
Ax + B
x2 + 1
+
C
x 1
+
D
(x 1)2
VillaRINO DoMath, FSMT-UPSI
Integration By Parts 19 / 20

20. Algebraic Procedures Integration By Parts Partial Fractions
Example
Z
2x + 4
(x2 + 1) (x 1)2
dx
VillaRINO DoMath, FSMT-UPSI
Integration By Parts 20 / 20