Problem 23 - Calculus for Biology and Medicine

Question

In Problems 23-26, complete the square in the denominator and evaluate the integral. $$ \int \frac{1}{x^{2}-2 x+2} d x $$

Step-by-Step Solution

Answer

The integral evaluates to \( \arctan(x-1) + C \).

Step 1: Identify the quadratic expression

The denominator of the integral is \( x^2 - 2x + 2 \). To complete the square, start by identifying the quadratic expression in the denominator, which is \( x^2 - 2x + 2 \).

Step 2: Complete the square

We complete the square for \( x^2 - 2x + 2 \) by transforming it into a perfect square trinomial. Start by adding and subtracting \( (\frac{-2}{2})^2 = (-1)^2 = 1 \) inside the expression: \( x^2 - 2x + 2 = (x^2 - 2x + 1) + 1 = (x-1)^2 + 1 \).

Step 3: Substitute into the integral

Rewrite the integral with the completed square in the denominator: \( \int \frac{1}{(x-1)^2 + 1} \, dx \). This is a standard form that will use the arctangent integration formula.

Step 4: Recognize the standard integral form

The integral is now \( \int \frac{1}{(x-a)^2 + b^2} \, dx \) where \( a = 1 \) and \( b = 1 \). The integral of \( \frac{1}{(x-a)^2 + b^2} \) is \( \frac{1}{b} \arctan\left(\frac{x-a}{b}\right) + C \).

Step 5: Evaluate the integral

Apply the formula from Step 4 to evaluate the integral: \( \int \frac{1}{(x-1)^2 + 1} \, dx = \arctan(x-1) + C \).

Key Concepts

Completing the square

Completing the square is a powerful algebraic method used to manipulate quadratic expressions into a more manageable form. This technique makes integration and solving equations easier, particularly when dealing with quadratic expressions.

The expression we worked with was \( x^2 - 2x + 2 \). To complete the square, we want to transform this into a perfect square trinomial, \((x-a)^2 + b^2\). First, take the coefficient of \(x\) (which is \(-2\) in this case), divide it by \(2\), and then square it. This gives \((-1)^2 = 1\). Now, reorganize the quadratic expression:

• Add and subtract \(1\) (the value we just computed) within the expression.
• The expression \(x^2 - 2x + 2\) becomes \((x^2 - 2x + 1) + 1\).
• This simplifies further to \((x-1)^2 + 1\), which reveals how the expression comprises a perfect square trinomial plus a constant.

This transformation to \((x-1)^2 + 1\) is central for simplifying integration problems.

Integral evaluation

Integral evaluation involves finding the antiderivative or the area under the curve for a mathematical function. Once the quadratic expression is converted via completing the square, it opens up the path for integral evaluation.

Using our transformed expression \((x-1)^2 + 1\), we substitute back into the integral:

• The integral becomes \( \int \frac{1}{(x-1)^2 + 1} \, dx \).
• This integral is recognized as one that can be solved using standard integral formulas, particularly those involving arctangents.

Evaluating this integral involves recognizing the structure that matches a known formula, specifically the arctangent formula. This identification is crucial as it simplifies the process significantly, allowing the direct application of existing rules to find the antiderivative.

Arctangent integration formula

The arctangent integration formula is a specific tool used to evaluate integrals of a particular form. It's applied when dealing with integrals that represent inverse trigonometric functions.

For integrals of the form \( \int \frac{1}{(x-a)^2 + b^2} \, dx \), the solution utilizes:

• The formula: \( \frac{1}{b} \arctan\left(\frac{x-a}{b}\right) + C \), where \( C \) represents the constant of integration.
• In our example, \(a = 1\) and \(b = 1\), which simplifies the solution to \( \arctan(x-1) + C \).

This formula is essential for solving integrals that appear in problems involving completed squares and quadratic functions. Mastering its application will ease the process of solving many standard calculus problems.