Problem 51 - Calculus Single Variable

Question

Calculate the given integral. $$ \int \frac{2}{\left(x^{2}+1\right)^{2}} d x $$

Step-by-Step Solution

Answer

The integral evaluates to \( \tan^{-1}(x) + \frac{x}{x^2+1} + C \).

Step 1: Recognize the Integral Type

The integral given is \( \int \frac{2}{(x^2+1)^2} dx \). Recognize that this integral can often be solved using trigonometric substitution or by recognizing a differentiation pattern.

Step 2: Choose Substitution Method

Let's choose the substitution \( x = \tan(\theta) \), which simplifies \( x^2+1 \) to \( \sec^2(\theta) \). Then, \( dx = \sec^2(\theta) d\theta \). Substitute these into the integral.

Step 3: Substitute and Simplify

Substitute into the integral: \( \int \frac{2}{(\tan^2(\theta)+1)^2} \sec^2(\theta) d\theta \). Since \( \tan^2(\theta)+1 = \sec^2(\theta) \), it simplifies to \( \int \frac{2}{\sec^4(\theta)} \sec^2(\theta) d\theta = \int 2 \cos^2(\theta) d\theta \).

Step 4: Simplify Further Using Trigonometric Identity

Use the trigonometric identity \( \cos^2(\theta) = \frac{1 + \cos(2\theta)}{2} \) to rewrite the integral: \( \int (1 + \cos(2\theta)) d\theta \). This simplifies to \( \int d\theta + \int \cos(2\theta) d\theta \).

Step 5: Integrate Each Part

Integrate each part separately: \[ \int d\theta = \theta \] \[ \int \cos(2\theta) d\theta = \frac{1}{2}\sin(2\theta) \].Thus, the integral becomes \( \theta + \frac{1}{2}\sin(2\theta) + C \), where \( C \) is the constant of integration.

Step 6: Back-Substitute \( \theta \) in terms of \( x \)

Return to \( x \) by substituting back \( \theta = \tan^{-1}(x) \). Also, calculate \( \sin(2\theta) \) using double angle formulas: \( \sin(2\theta) = 2\sin(\theta)\cos(\theta) = \frac{2x}{x^2+1} \).

Step 7: Write Final Solution

The final integrated result is:\( \tan^{-1}(x) + \frac{x}{x^2+1} + C \), where \( C \) is the constant of integration.

Key Concepts

Trigonometric Substitution

Trigonometric substitution is a handy tool when dealing with certain integrals, especially when you see expressions like \( x^2 + a^2 \). It's often used to simplify integrals that contain square roots or power terms such as \( x^2 \).
For example, in the original problem, we have \( x^2 + 1 \). By using \( x = \tan(\theta) \), this expression simplifies to \( \sec^2(\theta) \). This substitution transforms the integral into a form that's often easier to integrate.
Here’s what happens step-by-step:

• First, replace \( x \) with \( \tan(\theta) \) which leads to \( dx = \sec^2(\theta) d\theta \).
• The expression \( x^2 + 1 \) now becomes \( \sec^2(\theta) \).
• This transforms your integral friendlier to work with and leads to simpler trigonometric integrals that are often solvable using basic identities.

By converting variables, you tap into the elegant world of trigonometric functions that often unravel the complexities initially presented in algebraic forms.

Integration Techniques

When tackling integral calculus problems, knowing a variety of integration techniques can be a lifesaver. For this particular problem, we used trigonometric identities to simplify the integral expression further. The goal is to transform the integral into a recognizable and easier-to-handle form using known techniques.
Here's how it worked in this particular case:

• After substitution, the integral becomes \( \int 2 \cos^2(\theta) d\theta \).
• To integrate \( \cos^2(\theta) \), use the identity \( \cos^2(\theta) = \frac{1 + \cos(2\theta)}{2} \). This breaks down the integral into the sum of two simpler integrals.

The integration process involves simplifying compound expressions and dealing with them one piece at a time. By identifying recognizable patterns and using identities, these techniques turn challenging integrals into more manageable calculations. So, having these tools in your math toolbox allows for creative problem-solving capabilities.

Differentiation Patterns

While solving integration problems, it’s valuable to recognize differentiation patterns that suggest how an integral can be efficiently approached. Differentiation is, by nature, the inverse process of integration. If certain parts of an integral resemble derivatives you’ve encountered, you can often use this to your advantage.

• In our solution, after performing trigonometric substitution and simplifying the integral to forms involving \( \theta \), we arrive at simple terms like \( \int d\theta \) and \( \int \cos(2\theta) d\theta \).
• Recognizing them helps because you know their respective antiderivatives right away: \( \theta \) for the first and \( \frac{1}{2}\sin(2\theta) \) for the second.

Being able to spot differentiation patterns turns abstract integrals into friendly calculations. This trick is a fundamental part of calculus that ties the concept of derivatives tightly with integrals. Recognizing these patterns means leveraging your differentiation knowledge directly within integration work.