Problem 70 - Calculus Single Variable

Question

Calculate the given integral. $$ \int \frac{5 x^{2}+20 x-3}{x^{2}+4 x+13} d x $$

Step-by-Step Solution

Answer

The integral of \( \int \frac{5 x^{2}+20 x-3}{x^{2}+4 x+13} \, dx \) is \( 5x - \arctan \left( \frac{x+2}{3} \right) + C. \)

Step 1: Determine if substitution can be applied

Examine the integrand \( \frac{5x^2 + 20x - 3}{x^2 + 4x + 13} \) to see if substitution can simplify the integration. Since the numerator has a similar degree to the denominator, substitution is not directly applicable. Consider polynomial long division instead.

Step 2: Perform Polynomial Long Division

Divide \( 5x^2 + 20x - 3 \) by \( x^2 + 4x + 13 \) to separate the integrand into a sum with a polynomial and a proper rational expression. Long division gives a quotient of 5 and a remainder of 0, resulting in:\[ \frac{5x^2 + 20x - 3}{x^2 + 4x + 13} = 5 + \frac{-3}{x^2 + 4x + 13}.\]

Step 3: Integrate the Polynomial Term

Integrate the polynomial term from the result of division:\[ \int 5 \, dx = 5x.\]

Step 4: Integrate the Rational Term

Integrate the term \( \frac{-3}{x^2 + 4x + 13} \). To do this, complete the square in the denominator:\( x^2 + 4x + 13 = (x+2)^2 + 9. \)Rewrite the integral and use the arctangent formula:\[ \int \frac{-3}{(x+2)^2 + 9} \, dx = -3 \int \frac{1}{(x+2)^2 + 3^2} \, dx. \] Use the substitution \( u = x + 2 \) and \( du = dx \), and apply the integral formula for arctan:\[ = -3 \cdot \frac{1}{3} \arctan \left( \frac{u}{3} \right) + C = -\arctan \left( \frac{x+2}{3} \right) + C. \]

Step 5: Combine and Simplify

Combine the results from Step 3 and Step 4 to obtain the final integrated expression:\[ 5x - \arctan \left( \frac{x+2}{3} \right) + C, \] where \( C \) is the constant of integration.

Key Concepts

Polynomial Long Division

Polynomial long division is a method used to simplify complex algebraic expressions, particularly when you're faced with a rational expression where the degree of the numerator is greater than or equal to the degree of the denominator. In simpler terms, it's like long division used for numbers, but instead applied to polynomials.

To perform polynomial long division:

• Start by dividing the first term of the numerator by the first term of the denominator.
• Multiply the entire denominator by this new term and subtract the result from the numerator.
• Repeat the process with the new polynomial formed until the degree of the reminder is less than the denominator's degree.

In the given exercise, this process allowed us to express \( \frac{5x^2 + 20x - 3}{x^2 + 4x + 13} \) as a simpler expression: \( 5 + \frac{-3}{x^2 + 4x + 13} \). This simplification makes the integral much easier to handle.

Integration by Substitution

Integration by substitution is a technique used to simplify an integral by making a substitution, typically turning a complicated integral into a simpler one. It's especially useful when dealing with composite functions where a substitution can align the integral with known formulas or simpler patterns.

In the original exercise, although direct substitution wasn't applied to the first integral due to the expression's form, it was essential for later steps.

• When integrating the term \( \frac{-3}{(x + 2)^2 + 9} \), substitution of \( u = x + 2 \) simplified the expression.
• This substitution transformed \( dx \) to \( du \), facilitating the application of the arctangent integration formula.

Using substitution strategically eases complexities and reveals paths to solve challenging integrals.

Arctangent Integration Formula

The arctangent integration formula is crucial when dealing with expressions resembling the derivative of the inverse tangent function. The standard form you want to match is:

\[ \int \frac{1}{u^2 + a^2} \, du = \frac{1}{a} \arctan \left( \frac{u}{a} \right) + C \]

In our exercise, after completing the square and making the substitution \( u = x + 2 \), the integral \( \int \frac{-3}{(x+2)^2 + 9} \, dx \) was rewritten to fit the arctangent formula:

• Here, \( a = 3 \), making the integral \( -3 \int \frac{1}{u^2 + 3^2} \, du \).
• Applying the formula, the integral resolves to \( -\frac{1}{3} \arctan \left( \frac{u}{3} \right) \), simplifying further to \( -\arctan \left( \frac{x+2}{3} \right) \).

Understanding the arctangent integration formula allows you to solve integrals involving expressions with squared terms effectively.

Completing the Square

Completing the square is a powerful technique for rewriting quadratic expressions in a form that is easier to integrate or solve. It involves expressing a quadratic polynomial as a perfect square trinomial plus or minus a constant.

Steps to complete the square include:

• Given a quadratic \( ax^2 + bx + c \), first factor out \( a \) if it's not 1.
• Divide the linear term coefficient, \( b \), by 2, then square this result.
• Add and subtract the squared value inside the expression, balancing it if \( a eq 1 \).

For the given integral's denominator, \( x^2 + 4x + 13 \), completing the square led to \( (x+2)^2 + 9 \):

• Half of 4 is 2, squared gives 4. Added inside the bracket, it perfectly complements the squared term.

This transformation not only simplified the integration by aligning with the arctangent formula but also helped make sense of the integral format.