Problem 74 - Essential Calculus

Question

Verify the value of the definite integral, where \(n\) is a positive integer. $$ \int_{-\pi}^{\pi} x^{2} \cos n x d x=\frac{(-1)^{n} 4 \pi}{n^{2}} $$

Step-by-Step Solution

Answer

The definite integral value \(\int_{-\pi}^{\pi} x^{2} \cos n x dx\) equals \(\frac{(-1)^{n} 4 \pi}{n^{2}}\), thus verifying the original equation.

Step 1: Understand the integration by parts formula

The integration by parts formula is given by \(\int u v dx = u \int v dx - \int u' (\int v dx) dx\). We can choose \(u=x^2\) and \(v=\cos nx\) to proceed in order to calculate the integral.

Step 2: Differentiating and integrating \(u\) and \(v\)

Differentiate \(u\) and integrate \(v\) respectively to obtain the values of \(du\) and \(v\) which will be substituted in the above equation, which gives \(du = 2xdx\) and \(\int v dx = \int \cos nx dx = \frac{1}{n}\sin nx\).

Step 3: Substitute \(u\), \(v\), \(du\) and \(\int v dx\) into the formula and find the integral terms

The integration by parts formula \(u \int v dx - \int u' (\int v dx) dx\) becomes \(x^2 \frac{1}{n}\sin nx - \int 2x (\frac{1}{n}\sin nx) dx\). Evaluate the integral from \(-\pi\) to \(\pi\) and observe the property of the sine function over the period \(-\pi\) to \(\pi\).

Step 4: Evaluate the first term

The first term becomes zero because the integration from \(-\pi\) to \(\pi\) of \(x^2 \sin nx\), is zero as \(\sin nx\) is an odd function and the product of even function \(x^2\) and odd function \(\sin nx\) results in an odd function and integral of an odd function over a symmetric interval is 0.

Step 5: Evaluate the second term

The second term can be calculated using integration by parts twice and observing that \(\int_{-\pi}^{\pi}x \sin nx dx = 0\) (due to the sine function being odd and \(x\) also being odd, the product becomes even but the integral over a symmetric interval of such a function equals 0) and \(\int_{-\pi}^{\pi} \sin nx dx = 0\) (the integral of the sine function from \(-\pi\) to \(\pi\) equals 0).

Step 6: Compute the integral

Computing the integral gives the result \(\int_{-\pi}^{\pi} x^{2} \cos n x dx=\frac{(-1)^{n} 4 \pi}{n^{2}}\) as expected.

Key Concepts

Integration by Parts

Integration by Parts is a powerful technique used to evaluate integrals of products of functions. It's especially handy when one of the functions simplifies upon differentiation, while the other is easy to integrate. The formula for Integration by Parts is: \[ \int u \, dv = uv - \int v \, du \]where:

• \(u\) is a function that becomes simpler upon differentiation.
• \(dv\) is a function where its antiderivative \(v\) can be easily calculated.

In the exercise, we used \(u = x^2\) and \(dv = \cos(nx) dx\) because differentiating \(x^2\) yields \(2x\), simplifying the problem. Integrating \(\cos(nx)\) gives \(\frac{1}{n} \sin(nx)\). By substituting these into the integration by parts formula, we begin the process of solving the integral analytically.

Trigonometric Integration

When dealing with integrals involving trigonometric functions, various strategies can be implemented. Trigonometric Integration involves specific techniques or identities that simplify the process. Integration of functions such as \(\sin(nx)\) and \(\cos(nx)\) often appear in many problems and leveraging their properties is crucial.In the current problem, trigonometric identities come into play. The integral of sine and cosine over certain symmetric intervals can be zero, as these functions exhibit particular symmetric properties. For instance, integrating \(\sin(nx)\) over \(-\pi\) to \(\pi\) results in zero because the integral of a full period wave on symmetric limits cancels out the positive and negative parts. Understanding these principles helps in simplifying the integration process, as shown in the exercise where terms involving \(\sin(nx)\) became zero.

Even and Odd Functions

Understanding the properties of even and odd functions greatly aids in integration problems, especially over symmetric intervals. A function \(f(x)\) is considered even if it satisfies \(f(-x) = f(x)\) and it's odd if \(f(-x) = -f(x)\). These properties offer powerful shortcuts during integration.In our exercise:

• The function \(x^2\) is even because \(x^2 = (-x)^2\).
• The function \(\sin(nx)\) is odd since \(\sin(-nx) = -\sin(nx)\).

The product \(x^2 \sin(nx)\) becomes an odd function. When integrating an odd function over a symmetric interval like \(-\pi\) to \(\pi\), the integral is zero. This concept simplifies the computation since it eliminates terms that might appear complex to integrate directly.

Symmetric Interval Integration

Symmetric Interval Integration is a technique that takes advantage of the symmetry of the limits of integration, commonly used when the interval is of the form \([-a, a]\). This type of integration taps into the properties of even and odd functions.In the exercise, when we see the limits \(-\pi\) to \(\pi\), it signals an opportunity to apply symmetric interval properties:

• If a function is odd (like \(x^2 \sin(nx)\)), the integral over a symmetric interval is zero.
• Even functions, on the other hand, would yield a non-zero integral over such intervals.

Leveraging these characteristics allows us to quickly dismiss unnecessary calculations and reach the integral’s final value efficiently.