Question

In the following integrals express the sines and cosines in exponential form and then integrate to show that: $$ \int_{-\pi}^{\pi} \sin 3 x \cos 4 x d x=0 $$

Short answer

The integral \( \int_{-\pi}^{\pi} \sin 3 x \cos 4 x \, dx \) evaluates to zero.

Step-by-step solution

Express Sine and Cosine in Exponential Form

Use the exponential form of the sine and cosine functions: \( \sin(kx) = \frac{e^{ikx} - e^{-ikx}}{2i} \) and \( \cos(kx) = \frac{e^{ikx} + e^{-ikx}}{2} \).Apply these identities to \( \sin(3x) \) and \( \cos(4x) \):\( \sin(3x) = \frac{e^{i3x} - e^{-i3x}}{2i} \) and \( \cos(4x) = \frac{e^{i4x} + e^{-i4x}}{2} \).

Formulate the Integral

Substitute the exponential forms of sine and cosine into the integral:\(\int_{-\pi}^{\pi} \sin 3 x \cos 4 x \, dx = \int_{-\pi}^{\pi} \left( \frac{e^{i3x} - e^{-i3x}}{2i} \right) \left( \frac{e^{i4x} + e^{-i4x}}{2} \right) \, dx .\)Simplify the integral expression.

Simplify the Integral

Multiply and simplify the terms inside the integral:\(\int_{-\pi}^{\pi} \left( \frac{e^{i3x} - e^{-i3x}}{2i} \times \frac{e^{i4x} + e^{-i4x}}{2} \right) dx = \int_{-\pi}^{\pi} \frac{1}{4i} \left( e^{i7x} + e^{-ix} - e^{-i7x} - e^{ix} \right) dx.\)

Separate the Integral

Separate the integral into four parts:\(\frac{1}{4i} \left( \int_{-\pi}^{\pi} e^{i7x} dx + \int_{-\pi}^{\pi} e^{-ix} dx - \int_{-\pi}^{\pi} e^{-i7x} dx - \int_{-\pi}^{\pi} e^{ix} dx \right).\)

Integrate Each Part

Evaluate each of the integrals individually. All the integrals are of the form \( \int_{-\pi}^{\pi} e^{ikx} dx \), which equals zero for any non-zero integer \(k\):\(\int_{-\pi}^{\pi} e^{ikx} dx = 0\text{ for any }k eq 0.\)Since \(e^{i7x}\), \(e^{-ix}\), \(e^{-i7x}\), and \(e^{ix}\) all have non-zero exponents, each integral evaluates to zero. Therefore, the original integral sums to zero.

Conclusion

Combine the results to conclude that:\(\frac{1}{4i} \times (0 + 0 - 0 - 0) = 0.\)Thus, \( \int_{-\pi}^{\pi} \sin 3 x \cos 4 x \, dx = 0 \).

Key concepts

Exponential Form

The exponential form of trigonometric functions is a powerful tool in calculus and complex analysis.
By expressing functions like sine and cosine in their exponential forms, computations can become easier and more intuitive.

For sine and cosine, the exponential forms are:
\[\text{For sine: } \ \sin(kx) = \frac{e^{ikx} - e^{-ikx}}{2i} \] \[\text{For cosine: } \ \cos(kx) = \frac{e^{ikx} + e^{-ikx}}{2} \] Using these forms, we can transform trigonometric integrals into integrals involving exponential functions, which are often easier to handle, especially when integrating over symmetric intervals like \(-\pi \text{ to } \pi\).

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all values of the variable.
In this exercise, the relevant identities used are the exponential forms of sine and cosine.

These identities help in converting complex trigonometric integrals into simpler forms.

For example: \[\text{Using } \sin(3x) = \frac{e^{i3x} - e^{-i3x}}{2i} \text{ and } \cos(4x) = \frac{e^{i4x} + e^{-i4x}}{2},\] we can rework the integral \(\int_{-\pi}^{\pi} \sin 3x \cos 4x \, dx\) into an expression involving exponential terms.
Understanding and using these identities is crucial for solving integrals involving products of sines and cosines.

Complex Integration

Complex integration involves integrating functions of a complex variable.
In this problem, the integrals become complex due to the exponential terms.

When we substituted \(\sin(3x) \text{ and } \cos(4x)\) with their exponential forms, we ended up with terms like \(\int_{-\pi}^{\pi} e^{i7x} \, dx\) and \(\int_{-\pi}^{\pi} e^{-ix} \, dx.\)
Each of these integrals evaluates to zero when the exponent is non-zero.

The general rule for such integrals is: \[\text{For any integer } k eq 0, \int_{-\pi}^{\pi} e^{ikx} dx = 0\text{.} \] This is due to the orthogonality of exponential functions over symmetric intervals.

This orthogonality is a fundamental property in Fourier analysis and simplifies evaluating complex integrals.

Integration Techniques

Various integration techniques simplify the process of solving integrals.
In this exercise, after converting the trigonometric functions to their exponential forms, we used:

• **Substitution technique**: Substituted \(\sin(3x) \text{ and } \cos(4x)\) with their exponential counterparts.
• **Separation of integrals**: Split the integral into four parts involving different exponential terms.
• **Evaluation of integrals**: Used known results about the integrals of exponential functions.

These combined techniques made it possible to simplify and evaluate the original integral, ultimately demonstrating that it equals zero.
Mastering these techniques helps in tackling a wide range of integrals more efficiently.