Problem 22 - Calculus for Biology and Medicine

Question

Use the Table of Integrals to compute each integral after manipulating the integrand in a suitable way. $$ \int \frac{3}{x^{2}-4 x+5} d x $$

Step-by-Step Solution

Answer

The integral evaluates to \( 3\arctan(x-2) + C \).

Step 1 - Recognize the Standard Form

Observe the integrand \( \frac{3}{x^{2}-4x+5} \). Notice that it resembles the form of \( \frac{1}{ax^2 + bx + c} \), which indicates a potential arctangent substitution after completing the square for the quadratic expression in the denominator.

Step 2 - Complete the Square in the Denominator

Rewrite the quadratic expression \( x^2 - 4x + 5 \). Completing the square gives:\[ x^2 - 4x + 5 = (x^2 - 4x + 4) + 1 = (x - 2)^2 + 1 \]This transforms the integral into: \[ \int \frac{3}{(x - 2)^2 + 1} \, dx \]

Step 3 - Use the Arctangent Integral Formula

The integral now takes the form \( \frac{1}{(x - h)^2 + a^2} \), where the antiderivative is known to be \( \frac{1}{a} \arctan \left( \frac{x-h}{a} \right) + C \). Here, \( h = 2 \) and \( a = 1 \), applying this:\[ \int \frac{3}{(x-2)^2+1} \, dx = 3 \cdot \arctan(x-2) + C \]

Step 4 - Factor in the Constant Coefficient

Considering the integral's coefficient, the final expression becomes \[ 3 \cdot \arctan(x-2) + C \], where \( C \) is the constant of integration.

Key Concepts

Completing the Square

Completing the square is a powerful algebraic technique used to transform a quadratic expression into a perfect square trinomial plus or minus a constant. This technique involves three main steps:

• Start with the standard form of a quadratic: \( ax^2 + bx + c \).
• Transform it to the form \((x-h)^2 + k\), where \(h\) and \(k\) are constants.
• Use the perfect square \((x - \frac{b}{2a})^2 = x^2 - \frac{b}{a}x + \frac{b^2}{4a^2}\) to help with transformation.

For example, let's complete the square for the quadratic expression \(x^2 - 4x + 5\):- Divide the coefficient of the linear term (\(-4\)) by 2 to get \(-2\).- Square this result to obtain \(4\), and add to the \(x^2 - 4x\) part, resulting in \(x^2 - 4x + 4\).- However, since we added \(4\) inside the parenthesis, we need to subtract it outside, aligning perfectly with our original expression as \((x-2)^2+1\).
Completing the square not only simplifies integration problems, but also helps in solving equations and in understanding the geometrical interpretation of quadratics.

Integral Tables

Integral tables are pre-computed lists of integral forms that serve as a quick reference for solving integration problems. They provide solutions to common integral forms, often seen when dealing with polynomial, trigonometric, exponential, and logarithmic functions. Here is how you can effectively use them:

• Recognize and rewrite the integral function to match a form that appears in the table. This may involve algebraic manipulation, such as completing the square or using trigonometric identities.
• Identify the parameters of your integral and match them to the listed integral forms to get the antiderivative directly.
• Apply any additional constants or transformations required by the problem, such as multiplying by constants or shifting variables.

For example, in the exercise, after completing the square, the denominator transforms to \((x-2)^2+1\), which is recognizable in the integral tables under forms related to arctangents. This allows us to quickly find the antiderivative without the need for more complex calculations. Using these tables speeds up the problem-solving process, making them an invaluable tool in any math student's arsenal.

Arctangent Substitution

Arctangent substitution is particularly useful when integrating expressions involving quadratic denominators. After manipulating the integral into a form similar to \( \int \frac{1}{(x-h)^2 + a^2} \, dx \), one can easily apply the arctangent formula: \[\int \frac{1}{(x-h)^2 + a^2} \, dx = \frac{1}{a} \arctan\left( \frac{x-h}{a} \right) + C\]This technique includes the following steps:

• Ensure the quadratic in the denominator is in the correct form, often involving completing the square first.
• Identify \(h\) and \(a\) from your denominator, which in the exercise leads to \( h = 2 \) and \( a = 1 \).
• Substitute these values into the arctangent formula to obtain the antiderivative.

For example, after completing the square, our integral \( \int \frac{3}{(x-2)^2+1} \, dx \) becomes immediately solvable using the arctangent formula. Applying these constants results in the antiderivative \( 3 \cdot \arctan(x-2) + C \), where \( C \) is the integration constant. Such substitution methods are useful for integrals that, at first glance, might seem daunting or impossible. They transform complex problems into standard forms, making integration straightforward.