Problem 14 - Calculus: One and Several Variables

Question

Calculate.$$\int \frac{x^{5}}{x-2} d x$$.

Step-by-Step Solution

Answer

The short version of the answer to this integral is: \[ \int \frac{x^5}{x-2} dx = \frac{x^5}{5} + x^4 + \frac{x^4}{2} + \frac{4x^3}{3} + 4x^2 + 16x + C \]

Step 1: Perform polynomial long division

To begin, perform long division between the numerator (\(x^5\)) and the denominator (\(x - 2\)) to simplify the integrand: \[ \require{enclose} \begin{array}{c|cc cc} \multicolumn{1}{r}{x^4} & +2x^3 & +4x^2 & +8x & +16 \\ \cline{2-5} x - 2 & x^5 & 0 & 0 & 0 & 0 \\ \cline{2-2} \multicolumn{1}{r}{x^5} & -2x^4 \\ \cline{2-3} 0 & 2x^4 & 0 & \\ \cline{3-3} 0 & 0 & 0&\\ \end{array} \] So, after polynomial long division, we have: \[ \frac{x^5}{x - 2} = x^4 + 2x^3 + 4x^2 + 8x + 16 \] Now, rewrite the integral: \[ \int \frac{x^5}{x-2} dx = \int (x^4 + 2x^3 + 4x^2 + 8x + 16) dx \]

Step 2: Integrate term by term

Now that the integrand is simplified, integrate each term: \[ \int (x^4 + 2x^3 + 4x^2 + 8x + 16) dx = \int x^4 dx + \int 2x^3 dx + \int 4x^2 dx + \int 8x dx + \int 16 dx \]

Step 3: Apply the power rule

Apply the power rule of integration, which states that: \[ \int x^n dx = \frac{x^{n+1}}{n+1} + C \] where \(C\) is the constant of integration: \[ \int x^4 dx + \int 2x^3 dx + \int 4x^2 dx + \int 8x dx + \int 16 dx = \frac{x^5}{5} + C \]

Step 4: Combine constants and simplify

Combine all the integration results to obtain the final answer: \[ \frac{x^5}{5} + x^{4}+2\left(\frac{x^4}{4}\right)+4\left(\frac{x^3}{3}\right)+8\left(\frac{x^2}{2}\right)+16x+C=\frac{x^5}{5}+x^{4}+\frac{x^{4}}{2}+\frac{4x^3}{3}+4x^2+16x+C \] Thus, the integral of the given function is: \[ \int \frac{x^5}{x-2} dx = \frac{x^5}{5} + x^4 + \frac{x^4}{2} + \frac{4x^3}{3} + 4x^2 + 16x + C \]

Key Concepts

Polynomial Long Division

Polynomial long division is like regular division but applied to polynomials. It allows us to divide one polynomial by another until we can't continue further. This process is especially useful when simplifying complex expressions before integration.

In the exercise, we divided \(x^5\) by \(x - 2\). This resulted in a simplified polynomial of \(x^4 + 2x^3 + 4x^2 + 8x + 16\). This step is crucial as it transforms a potentially challenging integral into a series of simpler terms that can be individually integrated.

To perform polynomial long division:

• Align the terms of both the dividend and divisor in descending order of their degrees.
• Divide the leading term of the dividend by the leading term of the divisor.
• Multiply the entire divisor by this result and subtract from the dividend.
• Repeat these steps with the remainder until the degree of the remainder is less than the degree of the divisor.

This approach makes it feasible to handle polynomial expressions when performing integration or various algebraic manipulations.

Power Rule of Integration

The power rule of integration is a fundamental concept, integral to solving problems involving integrals of polynomial terms. It provides a straightforward formula to integrate expressions of the form \(x^n\).

In the given exercise, the power rule states that: \[\int x^n \, dx = \frac{x^{n+1}}{n+1} + C\]where \(C\) is the constant of integration.

The power rule is applied to each term of the polynomial separately, which simplifies the calculation. For example, when integrating \(x^4\), we apply the power rule to obtain \(\frac{x^5}{5}\). Similarly, \(2x^3\) becomes \(\frac{2x^4}{4}\) after applying this rule.

Key aspects to remember about the power rule:

• It applies only when \(n eq -1\).
• Each term of the polynomial is integrated independently.
• The increased power is one more than the original, divided by the new power.

Understanding this rule allows us to effortlessly compute various polynomial integrals.

Indefinite Integral

An indefinite integral, also known as an antiderivative, is the reverse operation of differentiation. It represents a family of functions and includes a constant \(C\) which distinguishes it from a definite integral.

In the context of the exercise, the indefinite integral is \(\int \frac{x^5}{x-2} \, dx\). This represents the general form of antiderivatives of \(\frac{x^5}{x-2}\), inclusive of all possible vertical shifts of the function.

When solving indefinite integrals:

• Ensure the expression is simplified, often necessitating techniques like polynomial long division.
• Utilize formulas such as the power rule to find the antiderivative.
• Remember to add \(C\), the constant of integration, to indicate the family of solutions.

This concept is essential across calculus, serving as the basis for finding functions given their rates of change.