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Review Problems Test 3 1206
I. Evaluate each of the following integrals:
a)
x ln xdx ∫
1/ 2
3/ 2
3/ 2 1/ 2 3/ 2 3/ 2
Let ln ,
ln ln 3 3 3 9
u x dv x
du v x x
x x x dx x x x C
∫
b)
2 x csc 3 xdx ∫
2 , csc , , cot
cot cot cot ln sin
u x dv x du dx v x
x x xdx x x x C
∫
c)
2
dx
∫
2
2 2
2
2
2 tan , 2sec
2sec 2sec sec ln sec tan
4 4 tan 2sec
ln 2
x dx d
d d d C
x x C
∫ ∫ ∫
x
2 4 + x
d)
tan x sec xdx ∫
3 2 2 2 2 2
4 2 2
2
4 2 5 3
tan sec sec tan (tan 1) sec
(tan tan ) sec
tan , sec
( ) tan tan 5 3
x x xdx x x xdx
x x xdx
u x du xdx
u u du x x C
∫ ∫
∫
∫
e)
2 2 2 2
dx dx
x x x x
∫ ∫
2
2 2 2
2
2sin , 2 cos
2 cos 2 cos 1 csc
8sin 4 4sin 16sin^ cos^8
cot 8 8
x dx d
d d d
x C C x
θ θ θ^ θ
∫ ∫ ∫
x 2
2 4 − x
f)
2
3 2
x x dx x x x
∫
2 2
3 2 2
2 2 2 2 2 2 2
3 2
2 2
3 2 3 2
3 2 3 2
x x x dx x
x x x Ax B Cx D Ax B x Cx D
x x x x
Ax Bx Ax B Cx D
x
x x x Ax Bx Ax B Cx D
x x x x A x B x A C B D
A B A C B D
∫
2 2 2 2 2 2
2 2
2 1 2
2 2
2
2 2 4
ln( 1) 3 tan see below 2 1
tan , sec
6sec 1 3 6 cos 6 (1 cos 2 ) 3 sin 2 sec 2 2
C D
x x dx dx dx dx x x x x
u x du xdx u du
x x x
dx x
x dx d
d d d C
−
−
∫ ∫ ∫ ∫
∫
∫
∫ ∫ ∫
1 1
2 2 2
2 1 1 2 2 2 2
3sin cos 3 tan 3 3 tan
1 1 1
ln( 1) 3 tan 3 tan ln( 1) 2 1 1 2 1
x x C x x
x x x
x x I x x x x x x x
− −
− −
h)
2
2
x x dx x x
∫
2
2 2
2 2 2
2 2 2
x x dx x x
x x A Bx C A x Bx C x
x x x x x x
x A A
x x Ax A Bx Cx Bx C
A B B B
B C C C
A C
I dx x
∫
∫ 2 2
2 1
ln 2 1 ln( 4) tan 2 2 2 2
x dx dx x x
x x x C
−
∫ ∫
II. Solve the differential equation:
2 xdy = x − 16 dx y ; (4) = 0
2
2
2 2
2 2
2 2 1
2 2 1
1
4sec , 4sec tan , 16 4 tan
4 tan 4sec tan 4 tan 4 (sec 1) 4sec
4 tan 4 4 tan 4 4
16 4 tan 4
0 0 4 tan 0
xdy x dx y
dy x
dx x
x y dx x dx d x x
d d d
x x C C
x y x C
C C
−
−
−
∫
∫ ∫ ∫
2 2 1
2 1
16 4 tan 4
16 4sec 4
x y x
or
x y x
−
−
2 2
2 2
lim tan 0
tan 0 lim (^1 )
sec ( ) 1 lim lim sec 1
x
x
x x
x x x x x x
x x
→∞
→∞
−
→∞ − →∞
e)
1
1
lim(ln )
x
x
x
−
→ 1 0
1 1 1 1 1 2
1 1
2
1 1
1 1
lim(ln ) 0
First find:
lim ln(ln )
lim( 1) ln(ln ) 0
ln(ln ) lim 1
ln (^ 1) lim lim (^1) ln
lim lim 0 (^1 1) ln 1 ln
lim lim(ln )
x x x x x x
x x
x x
x x
x
x
x x
x
x
x x x
x x
x
x x
x x x x
x
− → − → → →
→ →
→ →
→ →
− ⎝^ ⎠
1 0 1
x e
− = =
IV. Determine whether each of the following integrals converges or diverges
and if it converges find its value:
a)
2
1 2 2 ( 1)
x dx x −
∫
1 2 2 2 2 2 1 1
2
2 (^2 ) 2
lim lim ( 1) ( 1) 2( 1)
So the integral diverges.
Find the antiderivative: 1, 2
t t t
x x dx dx x x x
u x du xdx
x dx u du C C
x u^ x
→ +^ →+
−
b)
9
dx x x
∞
3/ 2 1/ 2
9 3/ 2^9 9
lim lim 2 lim
lim 0 9 3 3
t t (^) t
t t t
t
dx x dx x x (^) x
t
∞ − −
→∞ →∞ →∞
→∞
V. Use the table below to evaluate each of the following integrals:
3 2 8 x − 3 x dx
, x > 0
x x dx x x dx x x C
a b
2
cot
4 csc
x dx
− x
2 2
2 2 2
2
csc , csc cot , 2
csc cot 1 ln
csc 4 csc
1 2 4 csc ln 2 csc
u x du x xdx a
x x du a a u dx C
x x u a u a^ u
x C x
3
dx
x + x
