10.3 Trigonometric Integrals

Trigonometric integrals are a key part of integration techniques. They build on trig identities and inverse functions to solve complex integrals involving sine, cosine, tangent, and their reciprocals.

Mastering these methods expands your integration toolbox. You'll learn to simplify trig expressions, use substitution, and apply power-reducing formulas to tackle a wide range of integrals in calculus and beyond.

Trigonometric Identities

Fundamental trigonometric identities

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• Pythagorean identities express relationships between trigonometric functions
  • $\sin^2 \theta + \cos^2 \theta = 1$
  • $1 + \tan^2 \theta = \sec^2 \theta$
  • $1 + \cot^2 \theta = \csc^2 \theta$
• Half-angle formulas allow expressing trigonometric functions of half an angle in terms of the original angle
  • $\sin \frac{\theta}{2} = \pm \sqrt{\frac{1 - \cos \theta}{2}}$
  • $\cos \frac{\theta}{2} = \pm \sqrt{\frac{1 + \cos \theta}{2}}$
  • $\tan \frac{\theta}{2} = \frac{1 - \cos \theta}{\sin \theta} = \frac{\sin \theta}{1 + \cos \theta}$

Simplifying trigonometric expressions

• Power-reducing formulas help simplify expressions with higher powers of trigonometric functions
  • $\sin^2 \theta = \frac{1 - \cos 2\theta}{2}$
  • $\cos^2 \theta = \frac{1 + \cos 2\theta}{2}$
  • $\tan^2 \theta = \frac{1 - \cos 2\theta}{1 + \cos 2\theta}$
• Odd and even functions have specific properties that can be used to simplify expressions
  • Sine and tangent are odd functions: $\sin(-\theta) = -\sin \theta$, $\tan(-\theta) = -\tan \theta$
  • Cosine, secant, and cosecant are even functions: $\cos(-\theta) = \cos \theta$, $\sec(-\theta) = \sec \theta$, $\csc(-\theta) = \csc \theta$

Inverse Trigonometric Integrals

Integration techniques for secant and tangent

• Secant integrals can be solved using substitution or by expressing secant in terms of tangent
  • $\int \sec \theta \, d\theta = \ln |\sec \theta + \tan \theta| + C$
  • $\int \sec^3 \theta \, d\theta = \frac{1}{2} (\sec \theta \tan \theta + \ln |\sec \theta + \tan \theta|) + C$
• Tangent integrals can be solved by substitution or using the power-reducing formulas
  • $\int \tan \theta \, d\theta = \ln |\sec \theta| + C$
  • $\int \tan^2 \theta \, d\theta = \tan \theta - \theta + C$

Integration techniques for cosecant and cotangent

• Cosecant integrals can be solved using substitution or by expressing cosecant in terms of cotangent
  • $\int \csc \theta \, d\theta = \ln |\csc \theta - \cot \theta| + C$
  • $\int \csc^3 \theta \, d\theta = -\frac{1}{2} (\csc \theta \cot \theta + \ln |\csc \theta - \cot \theta|) + C$
• Cotangent integrals can be solved by substitution or using the power-reducing formulas
  • $\int \cot \theta \, d\theta = \ln |\sin \theta| + C$
  • $\int \cot^2 \theta \, d\theta = -\cot \theta - \theta + C$