Question

In the following integrals express the sines and cosines in exponential form and then integrate to show that: $$\int_{-\pi}^{\pi} \sin 2 x \cos 3 x d x=0$$

Short answer

The integral evaluates to zero due to the orthogonality of the exponential functions.

Step-by-step solution

Expressing Sine in Exponential Form

We use Euler's formula to express sine in exponential form. Euler's formula states: \( \sin(x) = \frac{e^{ix} - e^{-ix}}{2i} \). Therefore, \( \sin(2x) \) can be written as: \[ \sin(2x) = \frac{e^{i2x} - e^{-i2x}}{2i} \]

Expressing Cosine in Exponential Form

Next, we use Euler's formula to express cosine in exponential form. Euler's formula states: \( \cos(x) = \frac{e^{ix} + e^{-ix}}{2} \). Therefore, \( \cos(3x) \) can be written as: \[ \cos(3x) = \frac{e^{i3x} + e^{-i3x}}{2} \]

Substitute Expressions into Integrand

Substitute the exponential forms of \( \sin(2x) \) and \( \cos(3x) \) into the integral: \[ \int_{-\pi}^{\pi} \sin(2x) \cos(3x) dx = \int_{-\pi}^{\pi} \left( \frac{e^{i2x} - e^{-i2x}}{2i} \right) \left( \frac{e^{i3x} + e^{-i3x}}{2} \right) dx \]

Simplify the Integral

Expand the product inside the integral: \[ \int_{-\pi}^{\pi} \frac{(e^{i2x} - e^{-i2x})(e^{i3x} + e^{-i3x})}{4i} dx \] This simplifies to: \[ \int_{-\pi}^{\pi} \frac{e^{i5x} + e^{-ix} - e^{i x} - e^{-i5x}}{4i} dx \]

Integrate Each Term Separately

Evaluate the integral of each term separately: \[ \int_{-\pi}^{\pi} \frac{e^{i5x}}{4i} dx + \int_{-\pi}^{\pi} \frac{e^{-ix}}{4i} dx - \int_{-\pi}^{\pi} \frac{e^{ix}}{4i} dx - \int_{-\pi}^{\pi} \frac{e^{-i5x}}{4i} dx \]

Use the Orthogonality of Exponentials

Use the orthogonality property of the exponential functions: \[ \int_{-\pi}^{\pi} e^{inx} dx = 0 \text{ for any non-zero integer n} \] Since all the terms inside the integrals have non-zero multiples of x, the result of each integral will be zero.

Combine Results

Combine the results of each term: \[ \frac{1}{4i} (0 + 0 - 0 - 0) = 0 \] Therefore, \[ \int_{-\pi}^{\pi} \sin(2x) \cos(3x) dx = 0 \]

Key concepts

Euler's Formula

To solve complex integrals involving trigonometric functions, Euler's formula is incredibly helpful. Euler's formula connects complex exponentials to trigonometric functions:

• \( e^{ix} = \cos(x) + i\sin(x) \)
• \( e^{-ix} = \cos(x) - i\sin(x) \)

Using these relations, we can express sine and cosine in exponential form which makes it easier to handle the integrals. For instance:

• \( \sin(x) = \frac{e^{ix} - e^{-ix}}{2i} \)
• \( \cos(x) = \frac{e^{ix} + e^{-ix}}{2} \)

By converting trigonometric functions into exponential form, we reduce complex trig integrals into simpler exponential integrals.

Integral Calculus

Integral Calculus focuses on finding the integral of functions, which represents the area under their curves. In problems involving trigonometric integrals, the process often involves converting trigonometric expressions into forms that are straightforward to integrate.

• We express the trigonometric functions in their exponential forms.
• Next, we expand and simplify the integrand.
• Finally, we integrate each term separately within given limits.

For instance, in our problem, we converted \( \sin(2x) \) and \( \cos(3x) \) to their exponential forms, simplified the integral, and then tackled each exponential term separately to find the solution.

Orthogonality of Exponentials

One of the most beneficial properties when dealing with exponential functions in integrals is their orthogonality. This property simplifies the evaluation significantly, especially when the integrals are calculated over a complete period of the sine or cosine functions.

• A key formula here is: \( \int_{-\pi}^{\pi} e^{inx} dx = 0 \text{ for any non-zero integer n} \).

This implies that when integrating complex exponentials over a symmetric interval around zero, the terms will cancel each other out, resulting in zero. This orthogonality property was used in our solution to show that the integral of the product of \( \sin(2x) \) and \( \cos(3x) \) is zero.

Trigonometric Integrals

Trigonometric integrals involve the integration of expressions containing sine and cosine functions. These integrals are common in signal processing, physics, and engineering.

• They can be simplified using exponential forms of sine and cosine.
• This method leverages Euler’s formula.

For example, in our problem, we converted \( \sin(2x) \) to \( \frac{e^{i2x} - e^{-i2x}}{2i} \) and \( \cos(3x) \) to \( \frac{e^{i3x} + e^{-i3x}}{2} \). After expanding and integrating each term, we used orthogonality to conclude that the result of the integral is zero.