3.2 Trigonometric Integrals

Trigonometric integrals are a crucial part of calculus, involving the integration of sine, cosine, tangent, and secant functions. These integrals often require special techniques like power-reducing formulas, half-angle formulas, and product-to-sum conversions.

Mastering trigonometric integrals helps you solve complex problems in physics, engineering, and mathematics. You'll learn to use reduction formulas, handle different powers and products of trig functions, and apply various substitution methods to simplify and solve these integrals.

Trigonometric Integrals

Integration of sine and cosine products

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• Integrals involving products and powers of sine and cosine functions $\int \sin^m x \cos^n x \, dx$
  • For odd powers of sine or cosine, apply power-reducing formulas to decrease the exponent
    • $\sin^2 x = 1 - \cos^2 x$ (pythagorean identity)
    • $\cos^2 x = 1 - \sin^2 x$ (pythagorean identity)
  • For even powers of both sine and cosine, use half-angle formulas to simplify the integrand
    • $\sin^2 x = \frac{1 - \cos 2x}{2}$ expresses $\sin^2 x$ in terms of $\cos 2x$
    • $\cos^2 x = \frac{1 + \cos 2x}{2}$ expresses $\cos^2 x$ in terms of $\cos 2x$
  • After applying these formulas, integrate the simplified expression using substitution ($u$-substitution) or other integration techniques (integration by parts)
• Integrals involving products of sine and cosine with different arguments $\int \sin(mx) \cos(nx) \, dx$
  • Convert products of sine and cosine into sums using product-to-sum formulas
    • $\sin A \cos B = \frac{1}{2}[\sin(A+B) + \sin(A-B)]$ (sum of sines)
    • $\cos A \sin B = \frac{1}{2}[\sin(A+B) - \sin(A-B)]$ (difference of sines)
  • Integrate the resulting sum of sine functions using substitution or other appropriate methods ($u$-substitution, trigonometric substitution)
• These integrals often involve periodic functions, which repeat their values at regular intervals

Integrals with tangent and secant

• Integrals involving powers of tangent $\int \tan^n x \, dx$
  • For odd powers of tangent, substitute $u = \tan x$, then $du = \sec^2 x \, dx$ (trigonometric substitution)
  • For even powers of tangent, use the identity $\tan^2 x = \sec^2 x - 1$ to reduce the power and simplify the integrand (pythagorean identity)
• Integrals involving powers of secant $\int \sec^n x \, dx$
  • For odd powers of secant (except $n=1$), substitute $u = \sec x + \tan x$, then $du = \sec x \tan x + \sec^2 x \, dx$ (Weierstrass substitution)
  • For even powers of secant, use the identity $\sec^2 x = 1 + \tan^2 x$ to reduce the power and simplify the integrand (pythagorean identity)
• Integrals involving products of secant and tangent $\int \sec x \tan x \, dx$
  • Substitute $u = \sec x$, then $du = \sec x \tan x \, dx$ to simplify the integral ($u$-substitution)
• Remember to use radian measure when working with trigonometric functions in calculus

Reduction formulas for trigonometric integrals

• Reduction formula for integrals of sine raised to a power $\int \sin^n x \, dx$:
  • $\int \sin^n x \, dx = -\frac{1}{n} \sin^{n-1} x \cos x + \frac{n-1}{n} \int \sin^{n-2} x \, dx$ reduces the power of sine by 2 in each iteration
• Reduction formula for integrals of cosine raised to a power $\int \cos^n x \, dx$:
  • $\int \cos^n x \, dx = \frac{1}{n} \cos^{n-1} x \sin x + \frac{n-1}{n} \int \cos^{n-2} x \, dx$ reduces the power of cosine by 2 in each iteration
• Reduction formula for integrals of tangent raised to a power $\int \tan^n x \, dx$:
  • $\int \tan^n x \, dx = \frac{1}{n-1} \tan^{n-1} x - \int \tan^{n-2} x \, dx$ reduces the power of tangent by 1 in each iteration
• Reduction formula for integrals of secant raised to a power $\int \sec^n x \, dx$:
  • $\int \sec^n x \, dx = \frac{1}{n-1} \sec^{n-2} x \tan x + \frac{n-2}{n-1} \int \sec^{n-2} x \, dx$ reduces the power of secant by 2 in each iteration
• These formulas often rely on fundamental trigonometric identities to simplify expressions

Types of Trigonometric Integrals

• Antiderivatives: Indefinite integrals of trigonometric functions that result in a family of functions
• Definite integrals: Evaluate the area under a curve of a trigonometric function over a specific interval
• Indefinite integrals: Express the general antiderivative of a trigonometric function, including a constant of integration