Question

\(\pi \int_{-\pi}\left[\\{2 x(1+\sin x)\\} /\left(1+\cos ^{2} x\right)\right] d x\) is equal to....... (a) 0 (b) (c) \(\left(\pi^{2} / 2\right)\) (d) \(\pi^{2}\)

Short answer

(a) 0

Step-by-step solution

Simplifying the integrand

Let's first rewrite the integral: \(\pi\int_{-\pi}^{\pi}\frac{2x(1+\sin x)}{1+\cos^2 x}dx\) Our goal is to simplify it to make the integration process easier. Notice that \(1+\cos^2x\) can be written as \(\sin^2x + \cos^2x\) (since \(\sin^2x + \cos^2x = 1\)). Now, we can rewrite the integral as: \(\pi\int_{-\pi}^{\pi}\frac{2x(1+\sin x)}{\sin^2 x + \cos^2 x}dx\)

Observing the symmetry of the integrand

We will now observe the symmetry of the integrand \(\frac{2x(1+\sin x)}{\sin^2 x + \cos^2 x}\) The given integral is an even function with respect to x multiplied by an odd function of x (\(2x(1+\sin x)\)) and the denominator is always even. Now, the integrand becomes an odd function overall since: \(odd * even = odd\) Since the given function is odd, we may say that: \(\int_{-a}^{a}f(x) dx = 0\), for any odd function Hence, the integral evaluates to \(0\). So, \(\boxed{\text{(a) 0}}\) is the correct answer.

Key concepts

Symmetry in Integrals

Symmetry in definite integrals can simplify calculations and help us understand the properties of the function we're dealing with. Many integrals are symmetric, meaning that the function behaves in a predictable pattern around the y-axis. When examining \(\pi \int_{-\pi}^{\pi} \frac{2x(1+\sin x)}{1+\cos^2 x}dx\), symmetry plays a crucial role.
If an integrand is symmetric, the integral can sometimes be simplified or directly evaluated. The common types of symmetry are:

• Even Functions: Symmetrical about the y-axis, meaning that \(f(-x) = f(x)\). The integral of an even function over a symmetric interval sums up both sides equally.
• Odd Functions: Symmetrical about the origin, meaning that \(f(-x) = -f(x)\). Integrals of odd functions over symmetric intervals always equal zero because the areas on each side of the y-axis cancel each other out.

In our problem, noting that the integrand becomes an odd function overall helps us conclude the entire integral evaluates to zero. This is a powerful shortcut allowing us to bypass actual integration.

Odd and Even Functions

Understanding odd and even functions is essential when dealing with symmetry in integrals. This concept lets us predict the behavior of integrals without directly performing extensive calculations.
An even function satisfies the condition \(f(-x) = f(x)\). Visually, they are mirror images around the y-axis. Functions like \(x^2\) and \(\cos x\) are examples of even functions. When integrating even functions over a symmetric interval, we often simplify calculations since the symmetry allows us to focus on one side of the y-axis and double it.
On the other hand, an odd function satisfies \(f(-x) = -f(x)\). They are symmetric about the origin, meaning the left and right halves of the function are like mirror images flipped about the x-axis. Examples include \(x^3\) and \(\sin x\). When integrated over symmetric limits, odd functions result in an integral of zero, as the areas cancel each other out.
In our exercise, identifying the integrand as an odd function was key to quickly concluding that the integral equals zero. It’s a technique that can save considerable time and effort when properly applied.

Trigonometric Identities

Trigonometric identities can sometimes provide a useful way to simplify complex integral problems. In our example, the integrand involves the trigonometric functions \(\sin x\) and \(\cos x\), and simplifying them can often make the integral more approachable.
A commonly used identity is \(\sin^2 x + \cos^2 x = 1\). This identity shows the inherent relationship between sine and cosine, which can be beneficial for rewriting parts of an expression. In our case, it helped us reframe the denominator of our integrand, thereby possibly simplifying calculus steps.
Another important aspect of trigonometric identities involves expressing one trigonometric function in terms of another—such as expressing \(1 + \sin x\) or handling expressions involving \(1 + \cos^2 x\). These transformations can streamline calculations when integrating trigonometric expressions.
The ability to deftly use these identities makes tackling complex integrals more manageable, turning potentially tedious calculations into straightforward solutions.