Semester 2 Unit 5 –
5- 3
WKST
Name:
Integration by Parts. No Graphing Calculator is allowed for these problems.
For an integral,

f ( x) dx , such that
f ( x) dx can be written as u dv , then
 u dv  u v  v du .
The trick:
Let u = a power function . . . . unless . . . . if there exists a natural log function, then let u  ln x .
The Process:
Let u = one part of f ( x) dx
. . . . . . . . and let . . . . . . . . . . .
find du =
. . . . by taking the derivative, then take the integral to find . . . .
Plug in the above parts into the right side of the equation,
one more time and that’s it !!!!
Find the indefinite integral by Integration by parts.
1.
 xe
2.
x
2
x
dx
sin x dx
dv = the rest part of f ( x) dx
v=
 u dv  u v  v du , take the integral
3.
x
4.
 x sec
x dx
5.
xe
dx
10
ln x dx
2
3 x
ln x
6.

7.
 ln x dx
x10
dx
ANSWERS:
1) x e x  e x  C
2)  x 2 cos x  2 x sin x  2 cos x  C
3)
1 11
x ln
11
1 11
x  121
x C
4)  x tan x  ln cos x  C
5)  x3 e x  3x 2e x  6 x e x  6e x  C
6)  1 x 9 ln x  1 x 9  C
9
81
7) x ln x  x  C
8.
e
9.
xe
dx
Hint:
Let
x
cos x dx
3 x2
u  x2
and
2
dv  x e x dx
.