∫ sin^n x dx

Definition

The integral of the sine function raised to a positive integer power, with respect to the variable x. This type of integral is a fundamental concept in the study of trigonometric integrals, which are essential in various areas of calculus and mathematical analysis.

5 Must Know Facts For Your Next Test

1. The integral ∫ sin^n x dx is a special case of a trigonometric integral, where the integrand is a power of the sine function.
2. The value of the integral ∫ sin^n x dx depends on the value of the exponent n, and different techniques are used to evaluate the integral for different values of n.
3. When n is an odd positive integer, the integral can be evaluated using the substitution method, where a new variable is introduced to simplify the integration.
4. When n is an even positive integer, the integral can be evaluated using the integration by parts method, where the integral is split into two parts and then recombined.
5. The evaluation of ∫ sin^n x dx is often used in various applications, such as in the analysis of Fourier series and the solution of differential equations.

Review Questions

• Explain the general strategy for evaluating the integral ∫ sin^n x dx, where n is a positive integer.
  • The general strategy for evaluating the integral ∫ sin^n x dx, where n is a positive integer, depends on the parity of n. If n is odd, the substitution method can be used, where a new variable is introduced to simplify the integration. If n is even, the integration by parts method can be applied, where the integral is split into two parts and then recombined. Regardless of the value of n, the key is to identify the appropriate technique to transform the integral into a form that can be easily evaluated.
• Describe the steps involved in using the integration by parts method to evaluate the integral ∫ sin^2 x dx.
  • To evaluate the integral ∫ sin^2 x dx using the integration by parts method, the steps are as follows: 1) Let u = sin^2 x and dv = dx, so du = 2 sin x cos x dx and v = x. 2) Apply the integration by parts formula: ∫ u dv = uv - ∫ v du. 3) Substitute the values of u, dv, du, and v into the formula to get: ∫ sin^2 x dx = (x/2) sin^2 x - ∫ (x/2) 2 sin x cos x dx. 4) Recognize the remaining integral as ∫ sin x cos x dx, which can be further simplified using trigonometric identities.
• Analyze the relationship between the integral ∫ sin^n x dx and the Fourier series representation of periodic functions.
  • The integral ∫ sin^n x dx is closely related to the Fourier series representation of periodic functions. In Fourier analysis, the Fourier series expansion of a periodic function involves the integration of the function multiplied by sine and cosine terms. The coefficients of the sine and cosine terms in the Fourier series are directly related to the integrals of the function multiplied by the corresponding trigonometric functions. Therefore, the evaluation of ∫ sin^n x dx is crucial in determining the Fourier series representation of periodic functions, which has numerous applications in signal processing, electrical engineering, and other areas of mathematics and physics.

Related terms

Trigonometric Functions: The basic trigonometric functions, including sine, cosine, tangent, and their inverses, which describe the relationships between the sides and angles of right triangles.

Indefinite Integral: An antiderivative or primitive function of a given function, which represents the set of all functions whose derivative is the original function.

Integration by Parts: A technique used to evaluate integrals where the integral is divided into two parts, one of which is integrated and the other is differentiated, allowing for the original integral to be expressed in terms of a new integral.