Problem 63 - Calculus Single Variable

Question

Calculate the given integral. $$ \int \frac{4}{\left(x^{2}-2 x+2\right)^{2}} d x $$

Step-by-Step Solution

Answer

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

Step 1: Complete the Square

First, we need to simplify the expression inside the integral by completing the square in the denominator. The expression is \( x^2 - 2x + 2 \). Completing the square gives us \((x-1)^2 + 1\). Thus, the integral becomes \[\int \frac{4}{\left((x-1)^2 + 1\right)^2} \, dx\]

Step 2: Use Trigonometric Substitution

To simplify the integral, we use the substitution \( x - 1 = \tan(\theta) \). Then \( dx = \sec^2(\theta) \, d\theta \) and the integral becomes \[\int \frac{4}{(\tan^2(\theta) + 1)^2} \sec^2(\theta) \, d\theta\]Since \( \tan^2(\theta) + 1 = \sec^2(\theta) \), the integral simplifies to \[4 \int \cos^2(\theta) \, d\theta\]

Step 3: Use Double Angle Formula

To integrate \( \cos^2(\theta)\), we use the double angle formula: \[\cos^2(\theta) = \frac{1 + \cos(2\theta)}{2}\]Applying this identity results in the integral \[2 \int (1 + \cos(2\theta)) \, d\theta\] which simplifies to \[2 \left( \theta + \frac{1}{2}\sin(2\theta) \right) + C \]

Step 4: Back Substitute

Convert back to \( x \) by using the original substitution \( x - 1 = \tan(\theta) \), which implies \( \theta = \tan^{-1}(x - 1) \). Since \( \sin(2\theta) = 2\tan(\theta) \sec^2(\theta) \), we substitute back: \[2 \left( \tan^{-1}(x - 1) + \frac{1}{2} \cdot \frac{2(x-1)}{1 + (x-1)^2} \right) + C\] which simplifies to \[2 \tan^{-1}(x - 1) + \frac{2(x-1)}{x^2 - 2x + 2} + C\]

Key Concepts

Integration Techniques

Integration is a fundamental concept in calculus. It's often used to find areas under curves among other applications. There are various techniques to solve integrals, particularly those that are challenging or complex.
One common technique is **substitution**, where you change variables to simplify the integral. It works by replacing a part of the integrand with a single new variable.
Another technique is **partial fractions**, which is useful when the integrand is a rational function. This method involves breaking the function into simpler fractions.
For trigonometric integrals, specific formulas can simplify the process. The goal is to make the integrand easier to work with through these transformations.

Trigonometric Substitution

Trigonometric substitution is a technique used to simplify integrals involving radicals, especially when involving expressions like \(x^2 + a^2\), \(a^2 - x^2\), and \(x^2 - a^2\).

In our example, we used the substitution \(x - 1 = \tan(\theta)\). This transforms the integral into one involving trigonometric functions, which are often easier to handle.

To make the substitution, consider:

• **Set**: Identify the correct trigonometric identity to apply, matching the form of the expression with a trigonometric identity.
• **Transform**: Change the variable in the integral to a trigonometric function.
• **Simplify**: Use trigonometric identities to simplify the new integral.
• **Resubstitute**: Once completed, convert back to the original variable.

By converting using \(\sec^2(\theta) = \tan^2(\theta) + 1\), the integrand reduces considerably.

Completing the Square

Completing the square is an algebraic technique used to transform a quadratic expression into a perfect square trinomial.
In the integral problem, the expression in the denominator is \(x^2 - 2x + 2\).

To complete the square:

• First, **take half** the linear coefficient (-2) and square it, giving you 1.
• Then, **re-write** the expression as a perfect square: \((x-1)^2 + 1\).
• This transformation allows for easier substitution later in the integration process, making it more manageable.

Completing the square not only simplifies the functional form but can also reveal hidden symmetrical properties of the expression, paving the way for trigonometric substitutions.

Integral Simplification

Simplifying an integral makes it easier and quicker to solve. The process usually involves reducing the complexity of the integrand.
In our original integral example, the denominator was initially not in a simple conquering form.
By completing the square, we transformed a complex quadratic expression into a more straightforward expression suitable for substitution.
Similarly, by using trigonometric substitution and identities, such as \(\tan^2(\theta) + 1 = \sec^2(\theta)\), and the double angle formula for cosine \(\cos^2(\theta) = \frac{1 + \cos(2\theta)}{2}\), the integral is further simplified.
This process showcases how different methods work together. Each step reduces the problem into smaller, more manageable parts, allowing a comprehensive strategy for integration.