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"An Introduction to Integrals:
Techniques and Applications"
I
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g
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The Fundamental Theorem of Calculus

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Statement of the theorem
Proof of the theorem
Examples of using the theorem to evaluate definite integrals
III
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Integration by Parts

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Statement of the formula
Proof of the formula
Examples of using the formula to evaluate integrals
V
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g
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Exponential and Logarithmic Integrals

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Definition of the exponential and logarithmic functions
Theorems for integrating exponential and logarithmic functions (e
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∫a^x dx =
(a^x)/ln(a) + C)
Examples of using these theorems to evaluate integrals
VII
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g
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Hyperbolic Integrals

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Definition of the hyperbolic functions
Theorems for integrating hyperbolic functions (e
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∫cosh(x) dx = sinh(x) + C)
Examples of using these theorems to evaluate integrals
IX
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Applications of Integrals

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Finding areas
Solving differential equations
Other applications (e
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probability, physics)
XI
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Introduction:
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Definition of an integral: An integral is a mathematical operation that involves
evaluating a function over a specified domain and summing the resulting values
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Motivation for studying integrals: Integrals have a wide range of applications, including
finding the area bounded by a curve, calculating the volume of a solid, and solving
differential equations
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II
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Mathematically, this is
expressed as: ∫baf(x)dx = F(b) - F(a)
Proof of the theorem: The proof of the Fundamental Theorem of Calculus involves using
the definition of the integral and the definition of the derivative to show that the
theorem holds for any continuous function f(x)
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For example, to evaluate the integral ∫01x^2dx, we can find an antiderivative F(x) of x^2,
which is F(x) = (x^3)/3
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The Substitution Rule (continued):

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Statement of the rule: The Substitution Rule states that if u = g(x) is a differentiable
function and f is a continuous function, then the definite integral of f(u) with respect to u
can be expressed as an integral with respect to x: ∫bauf(u)du = ∫ba[f(g(x))g'(x)]dx
Proof of the rule: The proof of the Substitution Rule involves using the definition of the
integral and the chain rule to show that the rule holds for any differentiable function
g(x) and any continuous function f(u)
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For example, to evaluate the integral
∫0π/2sin(x^2)dx, we can make the substitution u = x^2, so that du = 2x dx
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Integration by Parts (continued):
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Statement of the formula: Integration by Parts states that if f and g are continuous
functions, then: ∫ba[f(x)g'(x)]dx = [f(x)g(x)]ba - ∫ba[f'(x)g(x)]dx
Proof of the formula: The proof of the formula for Integration by Parts involves using the
definition of the integral and the product rule to show that the formula holds for any
continuous functions f and g
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For example, to evaluate the integral ∫0π/2cos(x)sin(x)dx, we can use the
formula for integration by parts with f(x) = sin(x) and g'(x) = cos(x), to get:
∫0π/2cos(x)sin(x)dx = [sin(x)cos(x)]π/2 - ∫0π/2sin'(x)cos(x)dx The second integral on the
right-hand side can be evaluated using the Substitution Rule, with u = sin(x) and du =
cos(x) dx
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Trigonometric Integrals:

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Definition of the trigonometric functions: The trigonometric functions are a set of
functions that relate the angles of a right triangle to the lengths of its sides
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Theorems for integrating trigonometric functions: There are several theorems that can
be used to evaluate integrals involving trigonometric functions
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Examples of using these theorems: These theorems can be used to evaluate integrals
involving trigonometric functions
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Exponential and Logarithmic Integrals (continued):

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Definition of the exponential and logarithmic functions (continued): The logarithmic
function is the inverse of the exponential function, and is defined as y = loga(x), where a
is a positive constant and x is a positive number
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Some examples include: ∫a^x dx = (a^x)/ln(a) + C and ∫ln(ax) dx = xln(ax) - x
+ C
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For example, to evaluate the integral
∫10e^(2x)dx, we can use the formula ∫a^x dx = (a^x)/ln(a) + C with a = e^2 to get:
∫10e^(2x)dx = [(e^2)^x]/ln(e^2) + C = (e^2x)/2 + C
VII
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They allow us to solve for the angle
of a right triangle given the lengths of its sides
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Theorems for integrating inverse trigonometric functions: There are several theorems
that can be used to evaluate integrals involving inverse trigonometric functions
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Examples of using these theorems: These theorems can be used to evaluate integrals
involving inverse trigonometric functions
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Hyperbolic Integrals (continued):

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Definition of the hyperbolic functions (continued): The hyperbolic functions (continued) include the
hyperbolic cosine function (cosh), the hyperbolic tangent function (tanh), the hyperbolic cotangent
function (coth), the hyperbolic secant function (sech), and the hyperbolic cosecant function (csch)
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Some examples include: ∫cosh(x) dx = sinh(x) + C
and ∫sinh(x) dx = cosh(x) + C
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For example, to evaluate the integral ∫01sinh(x)dx, we can use the formula
∫sinh(x) dx = cosh(x) + C to get: ∫01sinh(x)dx = [cosh(x)]1 - [cosh(0)]0 + C = cosh(1) - 1 + C = (e 1)/2
IX
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Integration by partial fractions: Integration by partial fractions is a technique for evaluating integrals
of rational functions by expressing the rational function as a sum of simpler fractions that can be
integrated using other techniques
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X
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Solving differential equations: Integrals can be used to solve differential equations, by expressing the
solution as an indefinite integral that satisfies the given differential equation
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XI
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We covered techniques for
evaluating integrals, including the Fundamental Theorem of Calculus, the Substitution Rule,
Integration by Parts, and other techniques
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Future directions for study in the field of integrals: There are many advanced topics in the field of
integrals, including the evaluation of improper integrals, the use of numerical techniques to
approximate integrals, and the application of integrals to more advanced mathematical and scientific
concepts

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