Mathematics 53
Exercise Set 5
I. Evaluate the following limits. ln x 1−x √ 1. lim + x→1 x − 1 1− x 2.
6. lim+ sinh−1 z
3. lim
x→∞
x→∞
x
8.
x→0
II. Find
9. lim
x→0+
−x
cschx −
10.
lim
x→ 12 +
lim (tanh x + 1) √ x
x→∞
tan πx +
14. lim− x→1
sin 3πx cot πx
cschx
x→−∞
13. lim 1 x
x3
cosh x 1 1 + ex − e 1 − x
15. lim+ (cosh t − 1)
sinh t
x→0
dy . dx √
1. y =
2π
cosh x
√
+ (cosh x)
e
2. y = (πx) − (1 − cos x) 3. y = sinh−1 e + tan−1
√ 3
2π
12. y =
5
6. ln x + 1
y
+ tanh
3
13. y = √ √ 4 x coth 2x − cosh−1 x2
y x
−1
7. sinh−1 (ex+y ) + 3tan
x
−1
= coth √
= log
√ 15. y = ln cos tanh−1 (xy)
sech−1 x
Z 3. Z
√ 3
20. y =
Z 5.
1 + ln y cot (ln y) dy y ln y
6.
dν 1 + e2ν
7.
2
z e
z 3 +ez
3
dz
x3
x2 + 1 tan3 ex 18. y = sin (3 − 2x) √ p 19. y = (log3 x) cot−1 x csch−1 x + 2e2x
0
2.
sin−1 3−2x
√ 3
ecosh ex sinh πx − cosh πx
11. y = tan−1 (ln sinh x) + ln tan−1 (sinh x)
Z
√x
r √ 5 cos ln3 x 16. y = 3 tan−1 17. y = xx
9. 3x2 y + cosh2 (5x) = 2y
III. Evaluate the following integrals. Z 1. tan x sec (ln sec x) dx
sec−1 (2x) log3 (tanh x3 )
14. ln x2 + y 2 = sinh e−4x − 52y x3
2x + 1 x 8. y = ln 4 x−2
10. y =
2 x 3 ln cos x + esin(3 −x ) − tan 5x 4
x
p √ ln (cschx) − sin−1 x 4. y = sech−1 ln x √ √ log y 3 3 2 5 5. 7 7 − (csc x) = eπ + 3x y
4.
12.
2x+1
e − e − 2x x − sin x
5. lim
lim
x→+∞
2x − 3 2x + 5
x3 3 x +4
cot x
x→0
sin 3x2 4. lim x→0 ln cos (2x2 − x) x
11. lim
7. lim+ (cos x + 2 sin x)
lim (sec x − tan x) x 1+x
z→0
− x→ π 2
tan z
8.
ln 2
√ x3 + cos x + 23x · log e − x2 − 1 √2 xex2 + π e tan 5x
q
2e dq −q e 3−eq
Z
2 − πt dt eπt
Z
2 tanh2 2w + esinh 2w dw sech2w
Z
27u du 3u − 3
Z 9.
Z
e3v dv 2v e + 6ev + 9
21.
3 2 3log2 (χ +2χ ) √ 10. dχ log χ+2 1 χ9 2 Z dτ √ 11. τ −1 Z cosh (ln a) − coth (ln a) 12. da a sinh (ln a) Z 0 m 2 dm √ 13. 1 4 − 4m 2 Z √ s−1 √ ds 14. (s + 4) s Z p p 15. dp 2 p − 2p + 2 Z 3k + 2 √ dk 16. (3k − 2) 9k 2 − 12k − 5 Z csc2 φ tan φdφ p 17. csc4 φ − 2 csc2 φ − 8 Z dω 18. eω − ee−ω √ Z sinh 2θ √ dθ 19. 1 − sinh2 2θ Z 3 x −1 20. dx x2 + 1
Z
2
√
tan α tan2 α − 3
dα
Z
coth2 er dr cosh r − sinh r
Z
csch2 β coth β dβ 2 2csch β
22. 23.
csc2 γdγ
Z 24.
p Z
25.
csc2
tan7 FdF
Z 26.
γ − 4 cot γ − 6
2dη η
p
4η 4 − 5
Z
1
27. (ln x) Z
dη
ln
√
q
dx 2
(x ln x2 ) − x2
3
sechµ (2 − sinh µ tanh µ) dµ
28. 0
r2 − 1
Z 29.
(r4 + 3r2 + 1) tan−1 Z
2 tan ξ + 3 dξ sin2 ξ + 2 cos2 ξ
Z
3♣ + 1 d♣ 9♣ + 1
30. 31.
r 2 +1 r
dr
IV. Do as indicated. 1. Determine the equation of the tangent line to the graph of y = sech ln √ point where x = 2.
x √ 2
√ tanh x − 2 at the + csc−1 x2
2. Show that sin−1 (tanh x) = tan−1 (sinh x). p 3. Find the perimeter of the region bounded by the graphs of x = 4 − y 2 , y = −1, y = 1 and the y-axis. dy 4. Find the complete solution of = ex−y sinh ey . dx Z x t2 5. Find the extreme values of f (x) = x2 e−x and F (x) = e− 2 1 − t2 dt. 1
dy 6. Find the derivative of the following functions. dx Z ln x3 (a) y = et dt tanh x2
Z
π
(b) ex2
p
sinh t + cosh
−1
Z tdt +
tan−1
√
2y
ln (1 + z) dz = 0 0
7. The slope of the tangent line at any point with coordinate (x, y) on a curve is given by
2x + 1 . (tan y + 2y) (ex2 + ex − e)
If the curve passes through the origin, find the equation of the curve. 8. Set-up a definite integral that represents the volume of the solid formed when the region bounded by the graphs of y = ln x, y = ex , y = 1 and y = e is revolved about the y-axis. Z x 2 tan−1 z dz 9. Evaluate lim 0 √ . x→∞ 1 + x2
2
1 1 1 + + ··· + . n+1 n+2 n+n
Z
z
10. Calculate lim
n→∞
(z − t) sin t2 dt
1 . 12 Z 12. Compute value of the definite integral 11. Show that lim
z→0
0
ln (z 4 + 1)
=
π 4
sin θ + cos θ dθ 3 + sin 2θ 0 13. Find all functions f such that f 0 (x) = f (x) for all x. 14. Determine the sum Z
−5
−4
2
e(x+5) dx + 3
2 3
Z 1 3
2
e9(x− 3 ) dx 2