Question

Integrating \(\int \frac{P(x)}{Q(x)} d x\), where \(\operatorname{deg}(P(x)) \geq \operatorname{deg}(Q(x))\) Evaluate \(\int \frac{x^{2}+3 x+5}{x+1} d x\).

Short answer

\( \int \frac{x^2 + 3x + 5}{x+1} \, dx = \frac{x^2}{2} + 2x + 3\ln|x+1| + C \).

Step-by-step solution

Verify Degree Condition

First, check the degrees of the numerator and the denominator. The degree of the numerator \( P(x) = x^2 + 3x + 5 \) is 2, while the degree of the denominator \( Q(x) = x + 1 \) is 1. Since the degree of \( P(x) \) is greater than that of \( Q(x) \), polynomial long division is required.

Perform Polynomial Long Division

Divide \( x^2 + 3x + 5 \) by \( x + 1 \).1. The first term of the quotient is \( x \), since \( x \times (x+1) = x^2 + x \).2. Subtract \( x^2 + x \) from \( x^2 + 3x + 5 \) to get \( 2x + 5 \).3. The next term of the quotient is \( 2 \), since \( 2 \times (x+1) = 2x + 2 \).4. Subtract \( 2x + 2 \) from \( 2x + 5 \) to get the remainder \( 3 \).Thus, \( \frac{x^2 + 3x + 5}{x+1} = x + 2 + \frac{3}{x+1} \).

Integrate Each Part Separately

Integrate each term from the division result.1. Integrate \( x \), which gives \( \frac{x^2}{2} + C_1 \).2. Integrate \( 2 \), which gives \( 2x + C_2 \).3. Integrate \( \frac{3}{x+1} \), which results in \( 3\ln|x+1| + C_3 \).Combine these integrals: \( \frac{x^2}{2} + 2x + 3\ln|x+1| + C \), where \( C = C_1 + C_2 + C_3 \) represents the constant of integration.

Key concepts

Polynomial Long Division

When dealing with integrals where the degree of the polynomial in the numerator is equal to or greater than the degree of the polynomial in the denominator, polynomial long division comes into play. This technique is similar to the long division you might remember from arithmetic, but applied to polynomials.The main steps in polynomial long division are:

• Divide the highest degree term of the numerator by the highest degree term of the denominator. This gives you the first term of your quotient.
• Multiply the entire divisor by this term of the quotient and subtract the result from the original polynomial (or the remainder from the last step).
• Repeat the above steps with the new polynomial obtained by subtraction, until the degree of the remainder is less than the degree of the divisor.

In our original exercise, we divide \( x^2 + 3x + 5 \) by \( x + 1 \), which involves several iterations of division and subtraction until we reach a final remainder. The outcome is a polynomial quotient with a smaller remainder, ready for the next step in integration.

Integration Techniques

Once the polynomial long division is complete, the next step is integrating the resulting expression. This expression is typically composed of the quotient from the division and a fractional remainder.Here's how we manage each component:

• The terms of the polynomial quotient, such as \( x \) and \( 2 \) in our problem, are integrated directly using basic power rules. For example, \( \int x \) becomes \( \frac{x^2}{2} \), and \( \int 2 \) is \( 2x \).
• The remainder term, a simple rational expression like \( \frac{3}{x+1} \), is often a logarithmic integral, resulting from the integration of \( \int \frac{1}{x} \). For our case, it becomes \( 3\ln|x+1| \).

Integration requires attention to different types of expressions that emerge after polynomial division, and applying the appropriate technique ensures the final result is comprehensive and correct.

Degree of Polynomials

Understanding the degree of polynomials is crucial when tackling problems involving partial fraction decomposition and polynomial division. A polynomial's degree is determined by the highest power of the variable \( x \) in the expression.Important points to consider include:

• The degree of the numerator must be checked against the degree of the denominator. If the numerator's degree is greater than or equal to that of the denominator, long division is required to simplify the expression before integration.
• A correct assessment of degrees helps establish the correct strategy for solving integrals, as some may skip long division and use other techniques for integration directly when the numerator's degree is less than the denominator's.

By identifying the degrees early on, you can employ the right techniques and methods, ensuring a more streamlined and logical approach to both division and integration in polynomial calculus.