Math 9B Worksheet 5 Fall 2023
1. (5- pts) Integration by parts vs u-substitution (Remember LIATE rule when integrating
by parts)
(a) How can you tell the difference between a u-substitution integral and an integration
by parts integral?
(b) Evaluate
∫
t sec2(2t) dt.
(c) Evaluate
∫
t sec2 t tan t dt.
2. (5- pts) A wound is healing in such a way that t days since Sunday the area of the wound
has been decreasing at a rate of
t− 2t2
et−1
cm squared per day. This means
A(t) =
∫
t− 2t2
et−1
dt
The area of the wound on Monday was 8cm2, (i.e. A(1) = 8cm2).
(a) What was the area of the wound on Sunday? (Hint: Remember LIATE rule)
(b) What is the anticipated area of the wound on Thursday if it continues to heal at the
same rate?
(c) Use a calculator or computer to determine when the wound will be smaller than 1
square cm.
3. (optional extra practice) You don’t need to turn these in.
Evaluate
∫
sin(ln(x)) dx. Hint: Make a substitution t = ln(x), then integrate by parts.
4. (optional extra practice) You don’t need to turn these in. ˆ
∫
sin(ln(x)) dx ˆ
∫ 2
1
(lnx)2
x3
dx ˆ
∫
cos(
√
x) dx ˆ
∫
6 cos5 x sin4 x dx ˆ
∫ π/4
0
3 sin4 x cos2 x dx
∫ 3π/4
π/2
5 sin5 x cos3 x dx
∫
tan3 x secx dx 
∫ π/4
0
−3 sec4 x tan4 x dx
∫ √3
1
tan−1(1/x) dx