Problem 3 - Arbeitsbuch Mathematik für Ingenieure

Question

Berechnen Sie das uneigentliche Integral $$ \int_{-\infty}^{\infty} \frac{1}{\left(1+x^{2}\right)^{2}} d x $$

Step-by-Step Solution

Answer

The value of the integral is \(\frac{\pi}{2}\).

Step 1: Recognize the Integral Type

This is an improper integral because the limits of integration extend to \(-\infty\) and \(\infty\).

Step 2: Rewrite as Limit of a Definite Integral

Rewrite the integral using limits to handle the infinite bounds: \[ \int_{-\infty}^{\infty} \frac{1}{(1+x^{2})^{2}} dx = \lim_{a\to\infty} \int_{-a}^{a} \frac{1}{(1+x^{2})^{2}} dx \]

Step 3: Use Symmetry of the Function

Since the function \(\frac{1}{(1+x^{2})^{2}}\) is even, simplify the integral: \[ 2 \lim_{a\to\infty} \int_{0}^{a} \frac{1}{(1+x^{2})^{2}} dx \]

Step 4: Apply a Substitution

Use the substitution \(x = \tan\theta\) which gives \(dx = \sec^{2}\theta \, d\theta\). Then the new integral bounds are \(\theta = 0 \) to \(\theta = \tan^{-1}(a)\). After substitution, the integral becomes: \[ 2 \lim_{a\to\infty} \int_{0}^{tan^{-1}(a)} \frac{\sec^{2}\theta}{(1+\tan^{2}\theta)^{2}} d\theta \]

Step 5: Simplify the Trigonometric Integral

We know \(1+\tan^{2}\theta = \sec^{2}\theta\), so the integral simplifies to: \[ 2 \lim_{a\to\infty} \int_{0}^{\tan^{-1}(a)} \cos^{2}\theta \, d\theta \]

Step 6: Integrate and Evaluate the Limits

The integral of \(\cos^{2}\theta \) is evaluated using a half-angle identity: \[ \cos^{2}\theta = \frac{1+\cos(2\theta)}{2} \] Applying this identity and integrating: \[ 2 \lim_{a\to\infty} \left[\frac{1}{2}\theta + \frac{1}{4}\sin(2\theta)\right]_{0}^{\tan^{-1}(a)} \]

Step 7: Evaluate the Result

Plug in the bounds and simplify: \[ 2 \left( \lim_{a\to\infty} \left( \frac{1}{2} \tan^{-1}(a) + \frac{1}{4} \sin(2 \tan^{-1}(a)) \right) - 0 \right) \] As \(a \rightarrow \infty\), \(\tan^{-1}(a) \rightarrow \frac{\pi}{2}\) and \(\sin(2\tan^{-1}(a)) \rightarrow 0\), the expression becomes: \[ 2 \cdot \frac{1}{2} \cdot \frac{\pi}{2} = \frac{\pi}{2} \]

Key Concepts

Improper Integrals

Improper integrals are a type of integral where the limits of integration are infinite or where the function has an infinite discontinuity. These integrals are crucial in many areas of mathematics and physics. An example is the given integral, \( \int_{-\infty}^{\infty} \frac{1}{\left(1+x^{2}\right)^{2}} dx \). The limits of this integral extend from \( -\infty \) to \( \infty \), which makes it improper.
To solve improper integrals, we often rewrite them as the limit of a definite integral. For example, we rewrite the above integral as:
\[ \int_{-\infty}^{\infty} \frac{1}{(1+x^{2})^{2}} dx = \lim_{a\to\infty} \int_{-a}^{a} \frac{1}{(1+x^{2})^{2}} dx \] This allows us to handle the infinite bounds more easily and apply standard integration techniques.

Integration by Substitution

Integration by substitution is a powerful technique often used to simplify complex integrals. In essence, it involves substituting a part of the integrand (the function being integrated) with a new variable. This can transform a tricky integral into a more manageable one.
For our example, we use the substitution \( x = \tan\theta \). This choice is motivated by the form of the integrand and its relationship with basic trigonometric functions. The derivative of \( \tan\theta \) is \( \sec^{2}\theta d\theta \), hence the integral transforms to:
\[ \int \frac{1}{(1+x^{2})^{2}} dx = \int \frac{\sec^{2}\theta}{\sec^{4}\theta} d\theta = \int \cos^{2}\theta d\theta \] Here, the substitution significantly simplifies the original integral.

Trigonometric Integrals

Trigonometric integrals involve integrands that contain trigonometric functions like \( \sin x \), \( \cos x \), \( \tan x \), and others. Solving these integrals often requires identities and substitutions that leverage the properties of trigonometric functions.
In our example, after the substitution of \( x = \tan\theta \), we end up with the integral involving \( \cos^{2}\theta \). The next step is simplifying this trigonometric integral.
Using the identity \( \cos^{2}\theta = \frac{1 + \cos(2\theta)}{2} \), we convert the function into a form that is easy to integrate:
\[ \int \cos^{2}\theta d\theta = \int \frac{1 + \cos(2\theta)}{2} d\theta \] This identity breaks down the original function into manageable parts that can be integrated separately.

Half-Angle Identities

Half-angle identities are useful trigonometric identities that simplify the integration of trigonometric functions. They are particularly handy when dealing with integrals involving squared sine or cosine functions.
For example, the half-angle identity used in our solution is \( \cos^{2}\theta = \frac{1 + \cos(2\theta)}{2} \). This identity transforms the integral as follows:
\[ \int \cos^{2}\theta d\theta = \int \frac{1 + \cos(2\theta)}{2} d\theta = \frac{1}{2} \int (1 + \cos(2\theta)) d\theta \] By applying this identity, we can then integrate the two parts separately:

• \(\frac{1}{2} \int 1 \, d\theta = \frac{1}{2} \theta \)
• \(\frac{1}{2} \int \cos(2\theta) \, d\theta = \frac{1}{4} \sin(2\theta) \)

This simplification allows us to evaluate the integral efficiently, bringing us closer to the final solution. As we apply the established limits, we find that the original improper integral equals \( \frac{\pi}{2} \).