Question

Evaluate \(\int_{-3}^{3} \frac{x}{9-x^{2}} d x\) or show that it diverges.

Short answer

The integral evaluates to zero because the integrand is an odd function over a symmetric interval.

Step-by-step solution

Understand the Integral

The given integral to evaluate is \( \int_{-3}^{3} \frac{x}{9-x^{2}} \, dx \). We need to find the value of this definite integral or show that it diverges.

Identify Symmetry

Examine the integrand: \( f(x) = \frac{x}{9-x^{2}} \). Notice that it is an odd function because \( f(-x) = -f(x) \). Given that the limits of integration are symmetric around zero, \([-3, 3]\), we can conclude that the integral over a symmetric interval of an odd function is zero.

Analyze the Integrand

Consider the function more closely. \( 9-x^{2} \) equals zero when \( x = \pm 3 \), indicating that the integrand will be undefined (division by zero) at these endpoints. Direct evaluation at these points is needed to confirm any divergence.

Check for Convergence

Since the integrand becomes undefined at \( x = 3 \) and \( x = -3 \), the integral is improper. Therefore, interpret the integral as a limit: \[\lim_{a \to 3^-} \int_{-3}^{a} \frac{x}{9-x^{2}} \, dx + \lim_{b \to -3^+} \int_{b}^{3} \frac{x}{9-x^{2}} \, dx\]. However, both limits of the integrand and the domains symmetry around zero confirm that the integral evaluates to zero if handled symmetrically.

Key concepts

Improper Integral

An improper integral is a type of integral where the integrand becomes infinite or the limits of integration are infinite. In our given problem, we deal with a definite integral from \(-3\) to \(3\),where the integrand is \(\frac{x}{9-x^2}\). This function becomes undefined when \(x = \pm 3\) as the denominator reaches zero, causing a division by zero. To address improper integrals:

• Consider the behavior at the points where the function is undefined.
• Replace the problem point with a limit approaching the point from either the left or the right.

In this case, the limits become:\[\lim_{a \to 3^-} \int_{-3}^{a} \frac{x}{9-x^{2}} \, dx + \lim_{b \to -3^+} \int_{b}^{3} \frac{x}{9-x^{2}} \, dx\]This approach helps us address the undefined behavior while attempting to evaluate the integral's convergence or divergence.

Odd Function

Understanding the nature of the function involved can dramatically simplify evaluating integrals. An odd function is defined by the property \(f(-x) = -f(x)\). For the function \(f(x) = \frac{x}{9-x^2}\), this property holds as shown by:\[f(-x) = \frac{-x}{9-(-x)^2} = -\frac{x}{9-x^2} = -f(x)\]Odd functions have a special property when integrated over symmetric intervals: The integral of an odd function from \(-a\) to \(a\) is zero. Thus, the integral over the interval \([-3, 3]\) simplifies significantly and should generally yield zero unless there are discontinuities or undefined points which need specific attention.

Symmetric Interval

The term 'symmetric interval' refers to an interval that is symmetric with respect to the origin. In mathematical terms, intervals like \([-a, a]\) or \([-3, 3]\) are symmetric.In our problem, the interval under consideration is \([-3, 3]\).The symmetry can be very useful when integrating odd functions. When a function is odd, and the interval is symmetric around zero, the contribution of any function value at a point \(x\) cancels out with its function value at -\(x\). This arises from the property of odd functions:

• \(f(x) + f(-x) = 0\)

Thus, if the conditions of symmetry and oddness hold, the definite integral result will be zero.

Limits of Integration

In any definite integral, the limits of integration indicate the interval over which the function is being integrated, and they guide us in evaluating the integral.In this case, the limits are from \(-3\) to \(3\). However, due to the improper behavior at the endpoints (since the function is undefined at \(x = 3\) and \(x = -3\)), we handle these limits using limits in calculus.Using limits of integration helps:

• Address undefined points where the function approaches infinity.
• Ensure the process remains rigorous by avoiding direct substitution that leads to undefined expressions.

Considering these limits properly help clarify whether the integral converges to a finite number or diverges. Since symmetry and odd properties simplify our analysis, the improper interpretation confirms the result rather than complicating it.