Study Guide for Exam 1
I. Functions:
You must be able to do the following:
1. Graph the following functions and their families, and find, from the
graph, the limit, if exists, of any of these function at any point or as x
increases or decreases with no bound (i.e. the limits at infinity and minus
infinity):
A. The inverse trigonometric functions
B. The hyperbolic functions.
C. The inverse hyperbolic functions.
Examples:
Graph each of the following functions:
a. f(x) = 2 arcsin(3x)
b. f(x) = arctanh( x - 3 )
c. f(x) = - arctan(-x)
d. f(x) = - sechx
e. f(x) = coshx + 3
2. Find the value, if exists (must explain the reason if it does not) of any
of the six inverse trigonometric functions, the six hyperbolic functions at
a point and other values related to it.
Examples:
Find the value of each of the following:
a. arcsin(-1/2)
b. arcos( 3/2 )
e. arctccot (-√2)
f. cos[arctan(-5)]
g. sin[arccos(-1/5)]
h. arctan[cos(5π/4)]
i. arccos[tan(-π/4))]
j. cosh(ln5)
k. tanh(-5)
l. arcsec[sec(5π /4)]
m. arcsec[sec(- π /4)]
1
3. Prove all hyperbolic identities
Examples:
Prove each of the following:
a. cosh2x = sinh2x + 1
b. coth2x = cschh2x + 1
b. tanh2x + sech2x = 1
c. cosh2x = cosh2x +sinh2x
d. sinh2x = ( cosh2x – 1 ) / 2
4. Prove the logarithmic expressions for the inverse hyperbolic functions
II. Derivatives
You must be able to do the following:
1. Differentiate (Explicitly, implicitly and logarithmically) functions
involving all the studied functions (In the case of inverse hyperbolic
functions the formula will given)
Examples:
Differentiate each of the following functions
a. y = arctan8[ ln (cosx) ]
b. y = log3(arcsinx5)
c. y = 5arcsec(tan2x)
d. y = (arccos3x)sin5x
e. y = csc(e3x) arccotx7
f. y = (arccos3x)sin5x
g. y = [ 2 arctan5x + 8ln(tanx) ] / [ 2 arcsin5 + 2 arcsin(ex]
h. y = sinh8[ ln (cosx) ]
i. y = log3(coshx5)
j. y = 5tanh2x
k. y = e3x cschx7
l. y = (cosh3x)sin5x
m. y = [ 2 arctan(sihx6 )+ 8ln(coshx) ] / [ 2 coth5 + 2 coth(ex) ]
n. y = arcsec7 + x cosh 3
2
2. Prove the following formulas:
A. The derivatives formulas for the inverse trigonometric functions.
B. The derivative formulas for the hyperbolic functions.
C. The derivatives formulas for the inverse hyperbolic functions.
III. Integrals
You must be able to do the following:
1. Integrate by applying the appropriate integration formula.
Examples:
a. ∫ dx / x √ ( 25 – 4ln2 3x )
b. ∫ sec2( arcsin3x) dx / √ ( 1 - 9x2 )
c. ∫ x4 dx / (4 + 5 arcsinx5 ) √ ( 1 – x10 )
d. ∫ ln(arcsin2x) dx / √ ( 1 - 4x2 ) arcsin2x
e. ∫ 2x dx / (4x + 25 )
f. ∫ 2arctan5x dx / (1 + 25x2 )
g. ∫ 2x dx / [(4x + 1 ) arctan2x ]
h. ∫ 2x dx / [(4x + 1 ) arctan52x ])
i. ∫ cosh3x dx / √ ( 25 – 4sinh2 3x )
j. ∫ sech2( arcsin3x) dx / √ ( 1 - 9x2 )
k. ∫ sech2xdx / (tanh2x + 25 )
l. ∫sech5x thah5x 2sech5x dx
m. ∫ x2 tanh x3 dx
n. ∫x cschx2
o.
∫ 𝑑𝑥/√5 − 2𝑥 2
𝑑𝑥
p. ∫ 𝑥 2 −4𝑥−9
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2. Evaluate all of the following types of integrals, using the method of
integration by parts. No use of the reduction formulas or any ready
made formulas is accepted]
A. ∫xn cos(ax) dx, ∫xn sin(ax) dx, ∫xm eax dx , ∫xn cosh(ax) dx, ∫xn
sinh(ax) dx
where a is a constant
B. ∫xm ln(xn) dx, where n and m are real numbers
C. ∫ cos(ax) eax dx, ∫ cos(ax) eax dx, , ∫cos(bx)cosh(ax) dx,
∫sin(bx)sinh(ax) dx
D. ∫ x2n-1 cos(axn) eax dx, ∫ x2n-1 sin(axn),dx , ∫ x2n-1 eaxn
E. ∫arccosx dx, ∫arctanx dx, ∫arsecx dx, ∫ x arctanx dx and
similar integrals
F. ∫ secnx dx, ∫ cscnx dx where n is positive integer number greater
than 3.
G. ∫ tan−1 𝑎𝑥 𝑑𝑥, ∫ arcsin 𝑎𝑥 𝑑𝑥, ∫ tanh−1 𝑎𝑥 𝑑𝑥, ∫ sinh−1 𝑎𝑥 𝑑𝑥
H. ∫ sin (ln 𝑥) 𝑑𝑥
I. Find the reduction formulas for ∫ sinn 𝑥 𝑑𝑥, ∫ cos n 𝑥 𝑑𝑥,
∫ tann 𝑥 𝑑𝑥, ∫ sec n 𝑥 𝑑𝑥
3. Evaluate all of the following types of trigonometric and hyperbolic
Integrals from scratch. No use of the reduction formulas or any ready
made formulas is accepted]. Evaluate these integrals when x is replaced
by cx, for a constant c.
A. ∫sinmx cosnx dx, where either m or n is an odd natural number
B. ∫sinmx cosnx dx, where both m and n are even natural numbers
C. ∫secmx tannx dx, where n is an odd natural number
D. ∫secmx tannx dx, where m is an even natural number
E. ∫cscmx cotnx dx, where n is an odd natural number
F. ∫cscmx cotnx dx, where m is an even natural number
G. ∫sinhmx coshnx dx, where either m or n is an odd natural number
H.. ∫sinhmx coshnx dx, where both m and n are even natural numbers
I. ∫sechmx tanhnx dx, where n is an odd natural number
J. ∫sechmx tanhnx dx, where m is an even natural number
K. ∫cschmx cothnx dx, where n is an odd natural number
L.. ∫cschmx cothnx dx, where m is an even natural number
M. ∫sinax cosbx dx ; a≠ b
N. ∫sinax sinbx dx ; a≠ b
O. ∫cosax cosbx dx ; a≠ b
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3. Use the method of trigonometric substitution to evaluate:
A. Integrals, where the integrand contains an expression of the form ( ax2
+ b)n/2 , where n is an odd number and one or both of the constants a and
b being positive.
Examples:
Evaluate each of the following integrals
a.
x3

25  9 x
b.

c.
x
d.
x
e.
4 x 2  9  dx
x
1
2
25  9 x
2
1
2
9 x  25
2
.dx
.dx
9  4 x 2  dx
x2

f.

g

h.
.dx
2

x3
9  4x
x3
9  4x
2
2
.dx
.dx
dx
( x 2  6 x  15) 3
5
i.
j.


dx
( x 2  6 x  16)3
dx
(16  x 2  6 x)3
B. Integrals, where the integrand contains a quadratic expression of the
form cx2 +dx+ k.
Examples:
Evaluate each of the following integrals
a.
dx
 x2  4x  7
b.
dx
 1  x2  4x
c.
dx
 7  2x2  4x
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