91Ó°ÊÓ

Polynomial long division is a useful tool when dealing with integrals that involve rational functions. A rational function is one where you have a polynomial divided by another polynomial. In this case, we have the integral \( \int \frac{x^4}{x-1} \, dx \). To simplify it and make it easier to integrate, we use polynomial long division.

• First, view what you are dividing: the numerator \(x^4\) and the divisor \(x - 1\).
• Begin dividing as you would with numbers; divide the highest degree term of the numerator by the highest degree term of the divisor.
• The quotient from the first division is \(x^3\). Multiply this by the whole divisor \(x - 1\) to get \(x^4 - x^3\), then subtract this from the original polynomial.
• Continue the division process with the new polynomial \(x^3\). Repeat the steps until you arrive at a remainder, in this case, \(1\).

By the end, you've rewritten the function as \(x^3 + x^2 + x + 1 + \frac{1}{x-1}\), which is much easier to integrate.

An indefinite integral, unlike a definite integral, does not have upper and lower bounds. It represents a family of functions and includes a constant of integration, often denoted as \(C\). When dealing with polynomials, integrating each term is straightforward using basic integral rules:

• For \(x^n\), the integral is \(\int x^n \, dx = \frac{x^{n+1}}{n+1} + C\), where \(n\) is not \(-1\).

In our exercise, we are integrating the polynomial \(x^3 + x^2 + x + 1\). This gives us:

• \(\int x^3 \, dx = \frac{x^4}{4}\)
• \(\int x^2 \, dx = \frac{x^3}{3}\)
• \(\int x \, dx = \frac{x^2}{2}\)
• \(\int 1 \, dx = x\)

Put them all together to form the indefinite integral of the polynomial part as \(\frac{x^4}{4} + \frac{x^3}{3} + \frac{x^2}{2} + x\). Each term simplifies the integral and adds understanding to how different powers of \(x\) integrate.

Logarithmic integration comes into play particularly with the integration of expressions like \(\frac{1}{x-a}\). The integral of such a function results in a natural logarithm. This is because, for any integral \(\int \frac{1}{u} \, du\), the result is \(\ln |u| + C\), where \(C\) is the constant of integration. In the step-by-step solution:

• We specifically handle the fraction \(\frac{1}{x-1}\).
• Applying the integration rule \(\int \frac{1}{x-1} \, dx\), we arrive at the integral \(\ln |x-1| + C\).

This term captures the effect of the remainder from the polynomial long division and joins with the earlier polynomial integral. The full indefinite integral becomes \(\frac{x^4}{4} + \frac{x^3}{3} + \frac{x^2}{2} + x + \ln |x-1| + C\). Understanding logarithmic integrals is key as they frequently appear when dealing with rational functions, especially when those functions underscore a division by a linear term.