Lecture Notes on Exponential, Logarithmic, and Inverse Trigonometric Functions

Exponential Functions and Logarithms

Exponential functions with different bases can be expressed in terms of $e^x$ using logarithms. For example, $cos(x)^x = e^{x \cdot ln(cos(x))}$, assuming cos(x) > 0. The derivative of $a^x$ is $a^x \cdot ln(a)$, thus the integral of $a^x$ is $a^x / ln(a)$.

General Powers and Logarithmic Differentiation

For any real number $n$, the derivative of $x^n$ is $n \cdot x^{n-1}$. To differentiate functions like $y = x^{e^x}$, logarithmic differentiation can be used: take the logarithm of both sides, differentiate, and solve for $\frac{dy}{dx}$.

Logarithms with Different Bases

Logarithm in base $a$, denoted as $log<em>a(x)$, is the inverse function of $a^x$. It satisfies $a^y = x \iff log</em>a(x) = y$. Conversion formula: $log_a(x) = \frac{ln(x)}{ln(a)}$.

Inverse Trigonometric Functions

To define inverse trigonometric functions, we restrict the domains of the original trigonometric functions to make them one-to-one (injective).

• Arcsine (sin⁻¹(x) or arcsin(x)): Inverse of sine function restricted to $[-\frac{\pi}{2}, \frac{\pi}{2}]$. Domain: [-1, 1], Range: $[-\frac{\pi}{2}, \frac{\pi}{2}]$. Derivative: $\frac{d}{dx}arcsin(x) = \frac{1}{\sqrt{1-x^2}}$.

• Arccosine (cos⁻¹(x) or arccos(x)): Inverse of cosine function restricted to $[0, \pi]$. Domain: [-1, 1], Range: $[0, \pi]$. Derivative: $\frac{d}{dx}arccos(x) = -\frac{1}{\sqrt{1-x^2}}$.

• Arctangent (tan⁻¹(x) or arctan(x)): Inverse of tangent function restricted to $(-\frac{\pi}{2}, \frac{\pi}{2})$. Domain: all real numbers, Range: $(-\frac{\pi}{2}, \frac{\pi}{2})$. Derivative: \frac{d}{dx}arctan(x) = \frac{1}{1+x^2}}.

Applications of Inverse Trigonometric Functions in Integration

Inverse trigonometric functions provide antiderivatives for certain expressions. For instance:

$\int \frac{1}{\sqrt{1-x^2}} dx = arcsin(x) + C$

$\int \frac{1}{1+x^2} dx = arctan(x) + C$

When integrating expressions of the form $\int \frac{1}{\sqrt{a^2 - b^2x^2}} dx$, substitution (e.g., $u = \frac{3}{4}x$) can be used to reduce it to a standard form involving arcsine.