MAC2312
Quiz 1a
Z
π/2
1. Find
Name
ex cos x dx
0
Answer.
First find the indefinite integral, and then plug in the values at the end.
Integrate by parts, setting
u = cos x
dv = ex dx
du = − sin x v = ex
Then we get that
Z
x
x
Z
e cos x dx = e cos x +
ex sin x dx.
Now, do the same thing with the integral at the end, and get
Z
Z
x
x
e sin x dx = e sin x − ex cos x dx.
Now just put it all together:
Z
Z
x
x
x
x
e cos x dx = e cos x + e sin x − e cos x dx
Z
Z
x
x
x
e cos x dx = e cos x + e sin x −
ex cos x dx
|
{z
}
add this to both sides
Z
Z
x
e cos x dx + ex cos x dx = ex cos x + ex sin x
Z
2 ex sin x dx = ex sin x + ex cos x
Z
ex sin x + ex cos x
.
ex sin x dx =
2
Finally, plug in π/2 and 0 to get
eπ/2
−
2
1
.
2
Answer:
eπ/2 −1
2
MAC2312
Quiz 1a
Z
2. Find
Name
tan5 x sec x dx
Answer. Use the identities
d
dx
sec x = sec x tan x and tan2 x + 1 =
sec2 x. Rewrite the integral as
Z
Z
4
tan x sec x tan x dx = (sec2 −1)2 sec x tan x dx.
Then substitute u = sec x, du = sec x tan x dx. This gives:
Z
Z
2
2
(u − 1) du = (u4 − 2u2 + 1) du
= u5 /5 − 2u3 /3 + u + C
sec x(3 sec4 x − 10 sec2 x + 15)
=
.
15
Answer:
sec x(3 sec4 x−10 sec2 x+15)
15
+C
3. True or False?
a) If sec θ = 53 , then cot θ = 43 .
Answer: True
Z
b)
2
ex dx =
x2
e
2x
Answer: False