Problem 74 - Thomas Calculus

Question

Two complex numbers \(a+i b\) and \(c+i d\) are equal if and only if \(a=c\) and \(b=d .\) Use this fact to evaluate \begin{equation} \int e^{a x} \cos b x d x \quad \text { and } \int e^{a x} \sin b x d x \end{equation} from \begin{equation} \int e^{(a+i b) x} d x=\frac{a-i b}{a^{2}+b^{2}} e^{(a+i b) x}+C \end{equation} where \(C=C_{1}+i C_{2}\) is a complex constant of integration.

Step-by-Step Solution

Answer

The integrals are: \(\int e^{a x} \cos b x \, dx = \frac{a}{a^2+b^2} e^{a x} \cos b x + \frac{b}{a^2+b^2} e^{a x} \sin b x + C_1\) and \(\int e^{a x} \sin b x \, dx = \frac{b}{a^2+b^2} e^{a x} \cos b x - \frac{a}{a^2+b^2} e^{a x} \sin b x + C_2\).

Step 1: Rewrite the given complex integral solution

The given integral \( \int e^{(a+i b) x} d x \) results in \( \frac{a-i b}{a^{2}+b^{2}} e^{(a+i b) x} + C \). We will use this to find the real integrals \( \int e^{a x} \cos b x \ d x \) and \( \int e^{a x} \sin b x \ d x \) by equating the real and imaginary parts.

Step 2: Express \( e^{(a+i b) x} \) using Euler's formula

According to Euler's formula, \( e^{(a+i b) x} = e^{a x} e^{i b x} = e^{a x} (\cos b x + i \sin b x) \). By substituting this into the result of the integral of the complex exponential, we have: \[ \int e^{(a+i b)x} \, dx = \frac{a-i b}{a^2+b^2} e^{a x} (\cos b x + i \sin b x) + C. \]

Step 3: Separate into real and imaginary parts

The expression \( \frac{a-i b}{a^2+b^2} e^{a x} (\cos b x + i \sin b x) \) can be split into its real and imaginary components. The expression for \( e^{a x} (\cos b x + i \sin b x) \) implies that: \[ \int e^{(a+i b)x} \, dx = \Re \left(\int e^{a x} \cos b x \, dx\right) + i \Im \left(\int e^{a x} \sin b x \, dx\right) + C_1 + i C_2, \] where "\( \Re \)" and "\( \Im \)" denote the real and imaginary parts.

Step 4: Identify real integral expressions

By using equality of complex numbers, the real part corresponds to \( e^{a x} \cos b x \) and the imaginary part corresponds to \( e^{a x} \sin b x \). Thus, equating keeps us with: \[ \int e^{a x} \cos b x \, dx = \frac{a}{a^2 + b^2} e^{a x} \cos b x + \frac{b}{a^2 + b^2} e^{a x} \sin b x + C_1, \]\[ \int e^{a x} \sin b x \, dx = \frac{b}{a^2 + b^2} e^{a x} \cos b x - \frac{a}{a^2 + b^2} e^{a x} \sin b x + C_2.\]

Step 5: Write down the final expression

Summarizing the results from the previous step, we have: \( \int e^{a x} \cos b x \, dx = \frac{a}{a^2 + b^2} e^{a x} \cos b x + \frac{b}{a^2 + b^2} e^{a x} \sin b x + C_1 \) and \( \int e^{a x} \sin b x \, dx = \frac{b}{a^2 + b^2} e^{a x} \cos b x - \frac{a}{a^2 + b^2} e^{a x} \sin b x + C_2 \).

Key Concepts

Understanding Complex Numbers

Complex numbers are numbers that have two parts: a real part and an imaginary part. These numbers are usually written in the form \(a + ib\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit, which satisfies \(i^2 = -1\).

Here's why they matter:

• They allow us to solve equations that do not have real solutions, such as \(x^2 + 1 = 0\).
• Complex numbers are crucial in fields like electrical engineering and quantum physics.
• They enable more robust analysis of wave functions and oscillations.

In the context of integration, complex numbers help in combining oscillatory functions like sine and cosine into a single expression using Euler's formula.

That makes solving integrals involving exponential functions like \(e^{(a+ib)x}\) more manageable. You can separate an integrated expression into its real and imaginary parts, providing elegant solutions for cosines and sines.

Exploring Euler's Formula

Euler's formula is a pivotal link between complex numbers and trigonometry. It states that for any real number \(x\), \(e^{ix} = \cos x + i \sin x\). This means that complex exponentials can be expressed in terms of sine and cosine functions.
Euler's formula is beneficial because:

• It simplifies the multiplication and division of complex numbers, making operations more straightforward.
• It connects the exponential function with trigonometric functions in a seamless way.
• It reveals that exponential growth or decay can be oscillatory when the exponent is complex.

In integration problems, you can use Euler's formula to express \(e^{(a+ib)x}\) as \(e^{ax} (\cos(bx) + i \sin(bx))\).

This conversion is crucial for separating integrals into real and imaginary components, helping to identify solutions for real-life engineering problems, such as oscillations in circuits.

Real and Imaginary Parts in Integration

When working with complex numbers, it is often necessary to separate an expression into its real and imaginary parts. This separation is key in many fields of mathematics and engineering.
Here’s what you need to know:

• The real part refers to the component without the imaginary unit \(i\). For example, in \(a + ib\), \(a\) is the real part.
• The imaginary part includes the \(i\) term, with \(b\) being the coefficient in \(a + ib\).
• In integration, like \(\int e^{(a+ib)x} dx\), the resulting expression can be divided into real and imaginary parts to solve separately.

This technique was used in the exercise by equating the original complex integral's solution, separating it by Euler's identity.

Understanding real and imaginary parts allows us to deconvolute challenging expressions, providing analytical solutions for each component. This separation is a powerful tool when integrating functions involving complex exponentials like \(e^{ax} \cos bx\) and \(e^{ax} \sin bx\).