Question

Evaluate \(\int e^{(a+i b) x} d x\) and take real and imaginary parts to show that: $$\int e^{a x} \cos b x d x=\frac{e^{a x}(a \cos b x+b \sin b x)}{a^{2}+b^{2}}$$

Short answer

The real part of the integral is \( \frac{e^{a x} (a \cos b x + b \sin b x)}{a^2 + b^2} \).

Step-by-step solution

- Understand the integral

The goal is to evaluate the integral \(\int e^{(a+i b) x} d x \) and decompose it into its real and imaginary parts.

- Evaluate the complex integral

Rewrite the integral as follows: \( \int e^{(a+i b) x} d x = \int e^{a x} \cdot e^{i b x} d x \).Recognize that this is an integral of the form \( \int e^{(c) x} d x \) where \(c = a + ib\).

- Apply the integral

Use the basic integral rule for exponentials and evaluate the result: \( \int e^{c x} d x = \frac{e^{c x}}{c} + C \) \( \int e^{(a+i b) x} d x = \frac{e^{(a+i b) x}}{a+i b} + C \).

- Simplify the expression

Separate the real and imaginary components in the denominator: \[ \frac{1}{a+i b} = \frac{a-i b}{(a+i b)(a-i b)} = \frac{a-i b}{a^2 + b^2} \]. Thus, the integral can be written as: \[ \int e^{(a+i b) x} d x = e^{(a+i b) x} \cdot \frac{a-i b}{a^2 + b^2} + C \].

- Write in exponential terms

Split \(e^{(a+i b) x}\) into real and imaginary parts using Euler's formula: \( e^{(a+i b) x} = e^{a x} \cdot (\cos(b x) + i \sin(b x)) \).Substituting back in, the integral becomes: \[ \int e^{(a+i b) x} d x = \frac{e^{a x}(a \cos(b x) + a i \sin(b x) - i b \cos(b x) - b \sin(b x))}{a^2 + b^2} + C \].

- Identify real and imaginary parts

From the expression: \[ \frac{e^{a x}(a \cos(b x) + a i \sin(b x) - i b \cos(b x) - b \sin(b x))}{a^2 + b^2} + C \], identify the real part as \( \frac{e^{a x} (a \cos b x + b \sin b x)}{a^2 + b^2} \).

Key concepts

Complex Integration

Complex integration involves integrating functions that have complex numbers. Complex numbers include both real and imaginary parts, written as \(a + ib\), where \(a\) is the real part and \(ib\) represents the imaginary part. An integral like \(\int e^{(a+ib)x} dx\) combines these real and imaginary components.
To approach this, we first break down the exponential function using the properties of exponents: \(e^{(a+ib)x} = e^{ax} \times e^{ibx}\). Then, we use the basic rule for integrating exponentials: \(\int e^{cx} dx = \frac{e^{cx}}{c} + C\). Here, \(c = a + ib\).
Next, we simplify the fraction \(\frac{1}{a+ib}\) by multiplying by the conjugate: \(\frac{a-ib}{(a+ib)(a-ib)} = \frac{a-ib}{a^2+b^2}\). This helps us separate the real and imaginary parts more easily.

Euler's Formula

Euler's formula is a key tool in complex analysis. It states that \(e^{i\theta} = \cos \theta + i \sin \theta\). This formula allows us to express complex exponentials in terms of sine and cosine, which are easier to handle. When applying Euler's formula to our integral, we get: \(e^{(a+ib)x} = e^{ax} \times (\cos(bx) + i \sin(bx))\).
In this case, substituting back into our integral, we can more easily recognize the real and imaginary components. This simplification helps to directly identify the real part required in the problem statement.
With Euler's formula, we split our complex function such that we can apply real number operations to solve them.

Real and Imaginary Parts

To solve the integral involving complex numbers, it's crucial to separate the expression into its real and imaginary parts. Using Euler's formula, we can represent the integral result as: \(\frac{e^{ax}(a\cos(bx) + ai\sin(bx) - ib\cos(bx) - b\sin(bx))}{a^2+b^2} + C\).
By grouping the real parts together and the imaginary parts together, we identify the real component: \(\frac{e^{ax}(a \cos bx + b \sin bx)}{a^2 + b^2}\).
This method highlights how understanding each part of a complex function leads to determining the final real and imaginary components.