These integrals cover standard forms involving trigonometric functions like sin, cos, tan, and their powers. They are foundational for solving calculus problems involving periodic behavior, geometry, and waveforms.
Trigonometric Integrals
\[ \int \sin x \, dx = -\cos x + C \]
\[ \int \cos x \, dx = \sin x + C \]
\[ \int \tan x \, dx = \ln \sec x + C = -\ln \cos x + C \]
\[ \int \cot x \, dx = \ln \sin x + C \]
\[ \int \sin^2 x \, dx = \frac{x}{2} - \frac{1}{4} \sin 2x + C \]
\[ \int \cos^2 x \, dx = \frac{x}{2} + \frac{1}{4} \sin 2x + C \]
\[ \int \tan^2 x \, dx = \tan x - x + C \]
\[ \int \cot^2 x \, dx = -\cot x - x + C \]
\[ \int \sin^3 x \, dx = \frac{1}{3} \cos^3 x - \cos x + C \]
\[ \int \cos^3 x \, dx = \sin x - \frac{1}{3} \sin^3 x + C \]
\[ \int \frac{dx}{\sin^2 x} = -\cot x + C \]
\[ \int \frac{dx}{\cos^2 x} = \tan x + C \]
Terminology
Trigonometric Functions: Periodic functions such as \( \sin, \cos, \tan, \cot \).
Indefinite Integral: General form of an antiderivative with constant \( C \).
Trigonometric Identities: Used to simplify expressions for integration (e.g., power-reduction, double-angle formulas).
Power Integrals: Integrals involving powers of trigonometric functions (e.g., \( \sin^2 x, \cos^3 x \)).
Applications
Waveform analysis in electrical engineering and signal processing.
Calculating areas under trigonometric curves in physics and geometry.
Modeling harmonic motion and oscillation in mechanical systems.
Solving integrals in Fourier series, differential equations, and complex analysis.
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