Problem 42 - Essential Calculus

Question

Use a computer algebra system to find or evaluate the integral. $$ \int \frac{x^{2}}{x-1} d x $$

Step-by-Step Solution

Answer

The solution of the integral \( \int \frac {x^{2}}{x-1} dx \) is \( \frac {x^{2}}{2} + x + ln|x-1| + C \)

Step 1: Fraction Decomposition

Fraction decomposition is a method in which a complex fraction is decomposed into simpler fractions. Decompose \( \frac {x^{2}}{x-1} \) into \( x + 1 + \frac {1}{x-1} \).

Step 2: Integrate Each Part Separately

The integral of the function can be represented as the sum of the integrals of its parts. Integrate each part separately according to the power rule and the integral of a function defined as \( \frac {1}{x} \). Hence we have: \( \int x dx + \int dx + \int \frac {1}{x-1} dx \).

Step 3: Evaluate the Integrals

When evaluated, the integrals become \( \frac {x^{2}}{2} + x + ln|x-1| + C \), where C is the constant of integration.