MATH138: MIDTERM NOTES
𝑁(𝑥)
Rational Expression: 𝐷(𝑥)


Proper: if the highest exponent of x in N(x) is
less than the highest exponent of x in D(x).
Improper: if the highest exponent of x in N(x)
is equal to or greater than the highest
exponent of x in D(x).
Simplify by:
-
Division (long division)
Factoring
Integral Calculus (ANTIDIFFERENTIATION)
∫ 𝑓(𝑥) 𝑑𝑥 = 𝐹(𝑥) + 𝐶
 The elongated S is the symbol used in finding
the antiderivative or integral of a function. It is
called the integral sign.
 Integrand: It represents the function for which
the antiderivative satisfies. It is the derivative
of F(x).
 This is the differential. It “dictates” what is the
variable of integration. In this case, x is the
variable of integration. The variable in the
integrand must be consistent with the variable
of integration.
 This is the particular integral, the derivative of
which is equal to f(x).
𝑦 = 𝑓(𝑥)
y: dependent f(x): independent
 This is the Constant of Integration. It represents
any value, hence arbitrary.
 Integral calculus is the branch of Calculus
which deals with antidifferentiation, a process
used in solving areas of plane figures and
DEFINITE INTEGRALS
surfaces, volumes of solids, centroids,
moment of inertia, fluid pressure, work, and a
A definite integral is the value of an integral
lot more. It is a basic tool in solving most, if not
evaluated within specified boundaries. Generally, it
all, ENGINEERING problems.
represents the area bounded by the graph of a
function, 𝑓(x) , the x-axis and the lines x = a and x =
 Antidifferentiation (as the prefix implies) is the
b.
reverse process of differentiation.
Differentiation is the process of finding
derivatives of functions (whether algebraic of x = h (constant) : vertical lines
transcendental). Antidifferentiation, or simply
y = k (constant) : horizontal lines
INTEGRATION, is focused on obtaining the
functions given its derivative.
𝒙= 𝒃
∫
𝒇(𝒙) 𝒅𝒙 = [ 𝑭(𝒙) ]
𝒙=𝒂
𝒙 = 𝒃
𝒙 = 𝒂
𝒙= 𝒃
∫
𝒙=𝒂
𝒇(𝒙) 𝒅𝒙 = [𝑭(𝒙 = 𝒃)] − [ 𝑭(𝒙 = 𝒂)]
f(x) is a continuous real-valued function defined within
the closed interval [a,b] . x=a and x=b are called the
limits of integration. As shown in the figure, the
vertical lines x=a and x=b are boundaries of the
shaded area. F(x) is the particular integral to which
the values of the limits are applied
 A definite integral is a number or value. A
constant of integration is NOT added when
finding this value.
Finding Exact Differential (𝒅𝒖)
• Suppose 𝑢 = 3𝑥2 − 4𝑥 + 7. To find 𝒅𝒖,
differentiate 𝒖 with respect to 𝒙.
(correction factor & neutralizing factor)
BASIC FORMULAS IN GETTING ANTIDERIVATIVES
𝑭𝟏 : ∫ 𝒅𝒖 = 𝒖 + 𝑪
𝑭𝟗 : ∫ 𝐜𝐨𝐬 𝒖 𝒅𝒖 = 𝐬𝐢𝐧 𝒖 + 𝑪
𝑭𝟏𝟎 : ∫ 𝐭𝐚𝐧 𝒖 𝒅𝒖 = 𝒍𝒏 |𝐬𝐞𝐜 𝒖| + 𝑪
The integral of the exact differential of any
function is equal to the function itself plus 𝑪.
= − 𝒍𝒏 |𝐜𝐨𝐬 𝒖| + 𝑪
𝑭𝟏𝟏 : ∫ 𝐜𝐨𝐭 𝒖 𝒅𝒖 = 𝒍𝒏 |𝐬𝐢𝐧 𝒖| + 𝑪
𝑭𝟐 : ∫ 𝒂 𝒅𝒖 = 𝒂 ∫ 𝒅𝒖
where 𝒂 is any constant. If a differential is
multiplied by a constant 𝒂 (or if the integrand is a
constant 𝒂), the constant may be “factored out”
of the integral.
= − 𝒍𝒏 |𝐜𝐬𝐜 𝒖| + 𝑪
𝑭𝟏𝟐 : ∫ 𝐬𝐞𝐜 𝒖 𝒅𝒖 = 𝒍𝒏 |𝐬𝐞𝐜 𝒖 + 𝐭𝐚𝐧 𝒖| + 𝑪
𝑭𝟏𝟑 : ∫ 𝐜𝐬𝐜 𝒖 𝒅𝒖 = 𝒍𝒏 |𝐜𝐬𝐜 𝒖 − 𝐜𝐨𝐭 𝒖| + 𝑪
𝑭𝟑 : ∫[ 𝒇(𝒙) ± 𝒈(𝒙) ] 𝒅𝒙
= − 𝒍𝒏 |𝐜𝐬𝐜 𝒖 + 𝐜𝐨𝐭 𝒖| + 𝑪
= ∫ 𝒇(𝒙) 𝒅𝒙 ± ∫ 𝒈(𝒙) 𝒅𝒙 + 𝑪
𝑭𝟒 : ∫ 𝒖𝒏 𝒅𝒖 =
𝒖𝒏+𝟏
𝒏+𝟏
𝑭𝟏𝟓 : ∫ 𝒄𝒔𝒄𝟐 𝒖 𝒅𝒖 = − 𝐜𝐨𝐭 𝒖 + 𝑪
+𝑪
𝒘𝒉𝒆𝒓𝒆 𝒏 𝒊𝒔 𝒂𝒏𝒚 𝒏𝒖𝒎𝒃𝒆𝒓 𝒏 ≠ −𝟏. This is also
known as the Power Formula.
𝑭𝟓 : ∫
𝒅𝒖
𝒖
𝑭𝟏𝟒 : ∫ 𝒔𝒆𝒄𝟐 𝒖 𝒅𝒖 = 𝐭𝐚𝐧 𝒖 + 𝑪
= 𝒍𝒏 |𝒖| + 𝑪
𝑭𝟏𝟔 : ∫ 𝐬𝐞𝐜 𝒖 𝐭𝐚𝐧 𝒖 𝒅𝒖 = 𝐬𝐞𝐜 𝒖 + 𝑪
𝑭𝟏𝟕 : ∫ 𝐜𝐬𝐜 𝒖 𝐜𝐨𝐭 𝒖 𝒅𝒖 = − 𝐜𝐬𝐜 𝒖 + 𝑪
If n = -1 , 𝑭𝟓 is applicable if the “numerator” is
the exact differential of the “denominator”
Note: u = any function (the angle)
du = exact differential of u
INTEGRATION OF EXPONENTIAL FUNCTIONS
𝒖
𝑭𝟔 : ∫ 𝒂 𝒅𝒖 =
𝒂𝒖
𝒍𝒏 𝒂
+ 𝑪
𝒂 > 𝟎, 𝒂 ≠ 𝟏
𝑭𝟕 : ∫ 𝒆𝒖 𝒅𝒖 = 𝒆𝒖 + 𝑪
𝑭𝟖 : ∫ 𝐬𝐢𝐧 𝒖 𝒅𝒖 = − 𝐜𝐨𝐬 𝒖 + 𝑪
Trigonometric Identities
𝟐
𝟐
𝒔𝒊𝒏 𝒖 + 𝒄𝒐𝒔 𝒖 = 𝟏
𝟏 + 𝒕𝒂𝒏𝟐 𝒖 = 𝒔𝒆𝒄𝟐 𝒖
𝟏 + 𝒄𝒐𝒕𝟐 𝒖 = 𝒄𝒔𝒄𝟐 𝒖
𝐬𝐢𝐧(𝟐𝜽) = 𝟐 𝒔𝒊𝒏 𝜽 𝒄𝒐𝒔 𝜽
𝟏
𝒔𝒊𝒏(𝟐𝜽) = 𝒔𝒊𝒏 𝜽 𝒄𝒐𝒔 𝜽
𝟐
𝐜𝐨𝐬(𝟐𝜽) = 𝒄𝒐𝒔𝟐 𝜽 − 𝒔𝒊𝒏𝟐 𝜽
𝐭𝐚𝐧(𝟐𝜽) =
𝟐 𝒕𝒂𝒏 𝜽
𝟏 − 𝒕𝒂𝒏𝟐 𝜽
𝟏
𝟏
− 𝒄𝒐𝒔 𝟐𝒖
𝟐
𝟐
𝟏
𝟏
𝒄𝒐𝒔𝟐 𝒖 =
+ 𝒄𝒐𝒔 𝟐𝒖
𝟐
𝟐
𝒔𝒊𝒏𝟐 𝒖 =
𝒕𝒂𝒏𝟐 𝒖 =
𝟏 − 𝒄𝒐𝒔 (𝟐𝒖)
𝟏 + 𝒄𝒐𝒔 (𝟐𝒖)
∫ 𝟐 𝒔𝒊𝒏 𝒖 𝒄𝒐𝒔 𝒗 𝒅𝒙
= ∫[𝒔𝒊𝒏(𝒖 − 𝒗) + 𝒔𝒊𝒏(𝒖 + 𝒗)] 𝒅𝒙
∫ 𝟐 𝒄𝒐𝒔 𝒖 𝒄𝒐𝒔 𝒗 𝒅𝒙
= ∫[𝒄𝒐𝒔(𝒖 − 𝒗) + 𝒄𝒐𝒔(𝒖 + 𝒗)] 𝒅𝒙
∫ 𝟐 𝒔𝒊𝒏 𝒖 𝒔𝒊𝒏 𝒗 𝒅𝒙
= ∫[𝒄𝒐𝒔(𝒖 − 𝒗) − 𝒄𝒐𝒔(𝒖 + 𝒗)] 𝒅𝒙
∫ 𝒔𝒊𝒏 𝒖 𝒄𝒐𝒔 𝒗 𝒅𝒙
=
𝟏
𝟐
∫[𝒔𝒊𝒏(𝒖 − 𝒗) + 𝒔𝒊𝒏(𝒖 + 𝒗)] 𝒅𝒙
∫ 𝒄𝒐𝒔 𝒖 𝒄𝒐𝒔 𝒗 𝒅𝒙
=
𝟏
𝟐
∫[𝒄𝒐𝒔(𝒖 − 𝒗) + 𝒄𝒐𝒔(𝒖 + 𝒗)] 𝒅𝒙
∫ 𝒔𝒊𝒏 𝒖 𝒔𝒊𝒏 𝒗 𝒅𝒙
=
𝟏
𝟐
∫[𝒄𝒐𝒔(𝒖 − 𝒗) − 𝒄𝒐𝒔(𝒖 + 𝒗)] 𝒅𝒙
TRIGONOMETRIC TRANSFORMATION (POWERS of
SINE AND COSINE)
If the integrand involves powers of sine and
cosine, ∫ 𝒔𝒊𝒏𝒎 𝒖𝒄𝒐𝒔𝒏 𝒖 𝒅𝒖, this may be simplified
according to the following:
Case 1. When 𝑚 = 𝑛 = 1 𝑜𝑟, 𝑚 = 1 𝑎𝑛𝑑 𝑛 ≠ 1 𝑜𝑟 𝑚
≠ 1 𝑎𝑛𝑑 𝑛 = 1: evaluate the integral using
substitution.
Case 2. When 𝒎 is a positive ODD integer and 𝒏
is any number, write: 𝒔𝒊𝒏𝒎𝒖 𝒄𝒐𝒔𝒏𝒖 = 𝒔𝒊𝒏𝒎−𝟏𝒖
𝒄𝒐𝒔𝒏𝒖 (𝒔𝒊𝒏𝒖) du and apply Pythagorean
Identity 𝒔𝒊𝒏𝟐𝒖 = 𝟏 − 𝒄𝒐𝒔𝟐𝒖.
Case 3. When 𝒎 𝑖𝑠 𝑎𝑛𝑦 𝑛𝑢𝑚𝑏𝑒𝑟 𝑎𝑛𝑑 𝒏 𝑖𝑠 𝑎
𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒 𝑂𝐷𝐷 𝑖𝑛𝑡𝑒𝑔𝑒𝑟, write 𝒔𝒊𝒏𝒎𝒖 𝒄𝒐𝒔𝒏𝒖 =
𝒔𝒊𝒏𝒎𝒖 𝒄𝒐𝒔𝒏−𝟏𝒖 (𝒄𝒐𝒔𝒖) d𝒖 and apply
Pythagorean Identity 𝒄𝒐𝒔𝟐𝒖 = 𝟏 − 𝒔𝒊𝒏𝟐𝒖.
TRIGONOMETRIC TRANSFORMATION (POWERS of
TANGENT AND SECANT)
If the integrand involves powers of tangent and
secant: ∫ 𝒕𝒂𝒏𝒎 𝒖𝒔𝒆𝒄𝒖 𝒖 𝒅𝒖, the integrand may be
simplified according to the following:
Case 1: When m is any number and n is a
positive even integer greater than 2, write:
𝑡𝑎𝑛𝒎𝒖 𝑠𝑒𝑐 𝒏−2𝒖 (𝑠𝑒𝑐 2𝒖) and apply the identity:
𝒔𝒆𝒄𝟐𝒖 = 𝟏 + 𝒕𝒂𝒏𝟐𝒖.
Case 2: When 𝑚 is a positive ODD integer and 𝑛
Case 4: When 𝒎 and 𝒏 are both EVEN integers
(either both POSITIVE or one positive and the
other one zero), write:
𝒎
𝒏
𝒔𝒊𝒏 𝒖𝒄𝒐𝒔 𝒖 =
𝒎
𝒏
(𝒔𝒊𝒏𝟐 ) 𝟐 (𝒄𝒐𝒔𝟐 )𝟐
and
𝟏
(𝟏 − 𝒄𝒐𝒔𝟐𝒖)
𝟐
𝒄𝒐𝒔𝟐 𝒖 =
𝟏
𝟐
𝑠𝑒𝑐 𝒏−𝟏𝒖 (𝑠𝑒𝑐 𝒖 𝑡𝑎𝑛𝒖) and apply the identity:
𝑡𝑎𝑛𝟐𝒖 = 𝑠𝑒𝑐 𝟐𝒖 − 𝟏.
and apply
the identities:
𝒔𝒊𝒏𝟐 𝒖 =
is any number, write: 𝑡𝑎𝑛𝒎𝒖 𝑠𝑒𝑐 𝒏𝒖 = 𝑡𝑎𝑛𝒎−𝟏𝒖
(𝟏 + 𝒄𝒐𝒔𝟐𝒖)
Case 3. When 𝑚 is a positive odd or even
integer and 𝑛 is zero, write: 𝑡𝑎𝑛𝑚𝑢 = 𝑡𝑎𝑛𝑚−2𝑢
(𝑡𝑎𝑛2𝑢) and apply the identity:
𝑡𝑎𝑛2𝑢 = 𝑠𝑒𝑐2𝑢 − 1.
TRIGONOMETRIC TRANSFORMATION (POWERS of
COTANGENT AND COSECANT)
Case 1: When m is any number and n is a
positive even integer greater than 2, write:
𝒄𝒐𝒕𝒎𝒖 𝒄𝒔𝒄𝒏−2𝒖 (𝒄𝒔𝒄 2𝒖) and apply the identity:
𝒄𝒔𝒄𝟐𝒖 = 𝟏 + 𝒄𝒐𝒕𝟐𝒖.
Case 2: When 𝑚 is a positive ODD integer and 𝑛
is any number, write: 𝒄𝒐𝒕𝒎𝒖 𝒄𝒔𝒄𝒏𝒖 = 𝒄𝒐𝒕𝒎−𝟏𝒖
𝒄𝒔𝒄𝒏−𝟏𝒖 (𝒄𝒔𝒄𝒖 𝒄𝒐𝒕𝒖) and apply the identity:
𝒄𝒐𝒕𝟐𝒖 = 𝒄𝒔𝒄𝟐𝒖 − 𝟏.
METHODS OF INTEGRATION
It was shown in the previous topics how Formulas for
Integration are applied in finding the antiderivatives
of functions. It was also shown that some functions
need to be simplified before it can become
integrable.
There are, however, functions that require to be
simplified further using certain methods or
procedures. This chapter will discuss methods that
can be applied in simplifying integrands, depending
on the functions involved. Knowledge of these
methods enables one to derive additional formulas
for integration.
INTEGRATION BY PARTS
There are certain integrands that do not seem to fit
any of the standard formulas for integration.
Examples are: ∫ 𝒍𝒏 𝒙 𝒅𝒙 , ∫ 𝒂𝒓𝒄𝒕𝒂𝒏𝒙 𝒅𝒙, ∫ 𝒙√𝒙 + 𝟏 𝒅𝒙,
∫ 𝒆𝒙 𝐬𝐢𝐧 𝟐𝒙 𝒅𝒙, ∫ 𝒔𝒆𝒄𝟔 𝒙 𝒅𝒙 and more. Integration by
Parts can be used to simplify and render these
functions integrable.
Generally, Integration by Parts (IBP) is applicable if
the integrand involves:
1. Logarithmic functions
2. Inverse Trigonometric functions
3. Products such as:
a.) Exponential * Trigonometric
e.g ∫ 𝒆−𝒙 𝐬𝐢𝐧 𝟐𝒙 𝒅𝒙
b.) Exponential* Algebraic
e.g ∫ 𝒙𝟐 𝒆𝟑𝒙 𝒅𝒙
c.) Algebraic*Trigonometric
e.g ∫(𝒙 + 𝟏)𝟐 𝐜𝐨𝐬 𝒙 𝒅𝒙
d.) Algebraic*Trigonometric
e.g ∫(𝒙 + 𝟏)𝟐 𝐜𝐨𝐬 𝒙 𝒅𝒙
e.) Algebraic*Algebraic
e.g ∫ 𝒙√𝒙 + 𝟏 𝒅𝒙
4. Powers of Trigonometric Functions
Derivative Rules: