Problem 74 - Calculus for Biology and Medicine

Question

Evaluate each integral. $$ \int \frac{1}{x^{2}-x+2} d x $$

Step-by-Step Solution

Answer

The integral evaluates to \( \frac{2}{\sqrt{7}} \arctan\left(\frac{2x - 1}{\sqrt{7}}\right) + C \).

Step 1: Identify the Integral Type

The integral given is \( \int \frac{1}{x^2 - x + 2} \, dx \). It cannot be solved by direct integration since the denominator is a quadratic expression.

Step 2: Complete the Square

To simplify the expression, complete the square in the denominator. The expression \( x^2 - x + 2 \) can be rewritten as \( (x - \frac{1}{2})^2 + \frac{7}{4} \). This helps in transforming the integral into a form that is easier to solve.

Step 3: Recognize the Standard Form

The expression \( \frac{1}{(x-a)^2 + b^2} \) is now in the standard form for integration resulting in an arctangent function. The integral \( \int \frac{1}{(x-a)^2 + b^2} \, dx \) is \( \frac{1}{b} \arctan\left(\frac{x-a}{b}\right) + C \).

Step 4: Apply the Arctangent Formula

Using the formula from Step 3, substitute \( a = \frac{1}{2} \) and \( b = \sqrt{\frac{7}{4}} = \frac{\sqrt{7}}{2} \). The integral becomes \( \frac{2}{\sqrt{7}} \arctan\left(\frac{x - \frac{1}{2}}{\frac{\sqrt{7}}{2}}\right) + C \).

Step 5: Simplify the Result

Adjust the expression for clarity: \( \frac{2}{\sqrt{7}} \arctan\left(\frac{2x - 1}{\sqrt{7}}\right) + C \). This is the simplified form of the integral's solution.

Key Concepts

Completing the Square

Completing the square is a technique commonly used in algebra, particularly when dealing with quadratic expressions. It's a way to rewrite a quadratic expression in a form that is easier to handle, often turning it into a perfect square trinomial. Quadratic expressions generally take the form \( ax^2 + bx + c \). By completing the square, we transform it into \((x-h)^2 + k\), where \(h\) and \(k\) are constants that depend on the original coefficients \(a\), \(b\), and \(c\).
To complete the square, follow these steps:

• Identify the quadratic and linear coefficients from the expression.
• Divide the linear coefficient by 2, and square the result to find the number you need to complete the square.
• Rewrite the expression to include this perfect square trinomial and adjust any remaining constants.

For example, given \(x^2 - x + 2\), you can rearrange it by noting that \(-x\) divided by 2 is \(-\frac{1}{2}\), thus its square \(\left(-\frac{1}{2}\right)^2 = \frac{1}{4}\). So, \(x^2 - x + \frac{1}{4}\) forms \((x - \frac{1}{2})^2\), and along with the existing constant, it becomes \((x - \frac{1}{2})^2 + \frac{7}{4}\). This transformation is necessary for certain integration techniques, such as integration using the arctangent function.

Arctangent Integration

Arctangent integration is a powerful technique used when dealing with integrals that fit a specific form. Once you've completed the square for a quadratic expression in the denominator, you can often use arctangent integration. The standard form for such an integral is \(\int \frac{1}{(x-a)^2 + b^2} \, dx\), which results in the inverse trigonometric arctangent function.
Here are key steps for using arctangent integration:

• Ensure the quadratic expression is of the form \((x-a)^2 + b^2\).
• Recall the integral formula: \(\int \frac{1}{(x-a)^2 + b^2} \, dx = \frac{1}{b} \arctan\left(\frac{x-a}{b}\right) + C\).
• Identify \(a\) and \(b\) from your transformed expression after completing the square.
• Substitute these values into the arctangent formula to solve the integral.

For instance, let's say the square completed expression is \((x - \frac{1}{2})^2 + \frac{7}{4}\). Here, \(a=\frac{1}{2}\) and \(b=\frac{\sqrt{7}}{2}\). By substituting into the formula, you get \(\frac{2}{\sqrt{7}} \arctan\left(\frac{2x - 1}{\sqrt{7}}\right) + C\). This expression nicely simplifies the evaluation of the integral.

Quadratic Expressions

Quadratic expressions might seem daunting at first, but they form the basis of many fundamental concepts in mathematics, including integration. A quadratic expression has the general form \(ax^2 + bx + c\). These expressions can appear in various kinds of problems, especially in calculus when you encounter integrals involving them.
Dealing with quadratics involves understanding their properties and knowing techniques such as factorization or completing the square to transform them. Recognizing the need to manipulate these expressions often facilitates easier calculus operations like integration.
Some essential properties and methods related to quadratic expressions are:

• Graphing as a parabola, which can help visually understand roots and vertex.
• Completing the square, which transforms the expression into a readily usable form for certain calculations.
• Recognizing them within integrals to decide the appropriate method, like comparing to standard forms for integration involving inverse trigonometric functions.

Integration involving quadratics, especially in their incomplete forms, can be simplified using these fundamental techniques. Mastery comes through practice and familiarity with the transformations and results, like how completing the square played a key role in transforming the integral \(\int \frac{1}{x^2 - x + 2} \, dx\) in the original exercise.