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 loga(x), is the inverse function of a^x. It satisfies a^y = x \iff loga(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.