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Boundless Calculus
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Chapter 3

Inverse Functions and Advanced Integration

Book Version 1
By Boundless
Boundless Calculus
Calculus
by Boundless
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Section 1
Inverse Functions: Exponential, Logarithmic, and Trigonometric Functions
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Inverse Functions

An inverse function is a function that undoes another function.

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Derivatives of Exponential Functions

The derivative of the exponential function is equal to the value of the function.

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Logarithmic Functions

The logarithm of a number is the exponent by which another fixed value must be raised to produce that number.

Derivatives of Logarithmic Functions

The general form of the derivative of a logarithmic function is $\frac{d}{dx}\log_{b}(x) = \frac{1}{xln(b)}$.

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The Natural Logarithmic Function: Differentiation and Integration

Differentiation and integration of natural logarithms is based on the property $\frac{d}{dx}\ln(x) = \frac{1}{x}$.

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The Natural Exponential Function: Differentiation and Integration

The derivative of the exponential function $\frac{d}{dx}a^x = \ln(a)a^{x}$.

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Exponential Growth and Decay

Exponential growth occurs when the growth rate of the value of a mathematical function is proportional to the function's current value.

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Inverse Trigonometric Functions: Differentiation and Integration

It is useful to know the derivatives and antiderivatives of the inverse trigonometric functions.

Hyperbolic Functions

$\sinh$ and $\cosh$ are basic hyperbolic functions; $\sinh$ is defined as the following: $\sinh (x) = \frac{e^x - e^{-x}}{2}$.

Indeterminate Forms and L'Hôpital's Rule

Indeterminate forms like $\frac{0}{0}$ have no definite value; however, when a limit is indeterminate, l'Hôpital's rule can often be used to evaluate it.

Bases Other than e and their Applications

Among all choices for the base $b$, particularly common values for logarithms are $e$, $2$, and $10$.

Section 2
Techniques of Integration
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Basic Integration Principles

Integration is the process of finding the region bounded by a function; this process makes use of several important properties.

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Integration By Parts

Integration by parts is a way of integrating complex functions by breaking them down into separate parts and integrating them individually.

Trigonometric Integrals

The trigonometric integrals are a specific set of functions used to simplify complex mathematical expressions in order to evaluate them.

Trigonometric Substitution

Trigonometric functions can be substituted for other expressions to change the form of integrands and simplify the integration.

The Method of Partial Fractions

Partial fraction expansions provide an approach to integrating a general rational function.

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Integration Using Tables and Computers

Tables of known integrals or computer programs are commonly used for integration.

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Approximate Integration

The trapezoidal rule (also known as the trapezoid rule or trapezium rule) is a technique for approximating the definite integral $\int_{a}^{b} f(x)\, dx$.

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Improper Integrals

An Improper integral is the limit of a definite integral as an endpoint of the integral interval approaches either a real number or $\infty$ or $-\infty$.

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Numerical Integration

Numerical integration constitutes a broad family of algorithms for calculating the numerical value of a definite integral.

Section 3
Further Applications of Integration
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Arc Length and Surface Area

Infinitesimal calculus provides us general formulas for the arc length of a curve and the surface area of a solid.

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Area of a Surface of Revolution

If the curve is described by the function $y = f(x) (a≤x≤b)$, the area $A_y$ is given by the integral $A_x = 2\pi\int_a^bf(x)\sqrt{1+\left(f'(x)\right)^2} \, dx$ for revolution around the $x$-axis.

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Physics and Engineering: Fluid Pressure and Force

Pressure is given as $p = \frac{F}{A}$ or $p = \frac{dF_n}{dA}$, where $p$ is the pressure, $\mathbf{F}$ is the normal force, and $A$ is the area of the surface on contact.

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Physics and Engeineering: Center of Mass

For a continuous mass distribution, the position of center of mass is given as $\mathbf R = \frac 1M \int_V\rho(\mathbf{r}) \mathbf{r} dV$ .

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Applications to Economics and Biology

Calculus has broad applications in diverse fields of science; examples of integration can be found in economics and biology.

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Probability

Probability density function describes the relative likelihood, or probability, that a given variable will take on a value.

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Taylor Polynomials

A Taylor series is a representation of a function as an infinite sum of terms calculated from the values of the function's derivatives.

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Boundless Calculus by Boundless
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Chapter 2
Derivatives and Integrals
  • Derivatives
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Chapter 3
Inverse Functions and Advanced Integration
  • Inverse Functions: Exponential, Logarithmic, and Trigonometric Functions
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Differential Equations, Parametric Equations, and Sequences and Series
  • Differential Equations
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