Question

Derive the following formulas using the technique of integration by parts. Assume that n is a positive integer. These formulas are called reduction formulas because the exponent in the x term has been reduced by one in each case. The second integral is simpler than the original integral. $$ \int x^{n} \sin x d x= $$

Short answer

\(-x^n \cos x + n \int x^{n-1} \cos x \, dx\)."

Step-by-step solution

Identify Parts for Integration by Parts

In integration by parts, the formula \( \int u \, dv = uv - \int v \, du \) is used. Choose \( u = x^n \) and \( dv = \sin x \, dx \). This selection will make \( du = n x^{n-1} \, dx \) and \( v = -\cos x \) after integrating \( dv \).

Apply Integration by Parts

Substitute the chosen \( u \), \( dv \), \( du \), and \( v \) into the integration by parts formula: \[ \int x^n \sin x \, dx = -x^n \cos x + \int n x^{n-1} \cos x \, dx. \]

Simplify the Result

The expression \( -x^n \cos x + \int n x^{n-1} \cos x \, dx \) is the reduction formula because it reduces the exponent of \( x \) in the integral by 1. This second integral is simpler and can be handled using further integration techniques as necessary.

Key concepts

Reduction Formulas

Reduction formulas are a powerful tool when dealing with complex integrals involving powers of a variable. In this context, we use integration by parts to simplify the integral by reducing the exponent of a term. The key idea here is to transform a complicated integral into a simpler one.

For example, if we have an integral involving a term like \( x^n \sin x \), the goal is to reduce the exponent \( n \) of \( x \) through integration by parts. By strategically choosing functions \( u \) and \( dv \) in the integration by parts formula, we can create an expression where the exponent on \( x \) is decreased by 1 in a new, simpler integral.

In the original example, choosing \( u = x^n \) resulted in an expression where the next integral contained \( x^{n-1} \), illustrating how reduction formulas work. These formulas are particularly useful in integrating products of polynomial and trigonometric functions.

Trigonometric Integration

Trigonometric integration is the process of integrating functions that involve trigonometric expressions such as \( \sin x \), \( \cos x \), and others. These functions appear frequently in calculus, especially in problems involving periodic phenomena.

One common approach is using integration by parts, as shown in the exercise. By selecting a trigonometric function like \( \sin x \) as \( dv \), we can derive expressions for \( v \) by integrating, yielding results like \( v = -\cos x \). This technique is particularly useful when dealing with products of algebraic and trigonometric expressions—allowing us to systematically reduce the problem's complexity.

Employing integration techniques with trigonometric functions often involves repetitive application of integration by parts or using reduction formulas, progressively simplifying the integrals at each step.

Exponent Reduction

Exponent reduction is closely tied to the idea of simplifying integrals, where we focus on lowering the powers of terms to make the integral easier to evaluate.

In many integrals, particularly those involving polynomial expressions like \( x^n \), high exponents make direct integration challenging. By determining an optimal choice for functions in the integration by parts formula, such as setting \( u = x^n \), we can construct an integral that features \( x^{n-1} \).

This method supports the creation of a sequence of simpler integrals until reaching a point where the integrated term becomes a constant or a much simpler function. The primary purpose of exponent reduction is to break down difficult calculus problems into handleable steps, gradually addressing each subpart of the original integral.