Problem 61 - Calculus of a Single Variable

Question

Using Technology to Find an Integral In Exercises \(57-62\) 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 result of the integral \( \int \frac{x^{2}}{x-1} \, dx \) is \( \frac{1}{2}x^2 + x + C \).

Step 1: Divide the Integral Using Polynomial Long Division

Begin the task by dividing \(x^2\) by \(x - 1\). This can be done manually or using a software tool. The result of this operation is \(x + 1\).

Step 2: Reformulate the Integral

Now that the fraction has been divided, reformulate the integral as follows: \( \int (x + 1) \, dx \).

Step 3: Integrate the Simplified Formula

Now, you can integrate the simplified function. The integral of \(x\) is \( \frac{1}{2}x^2\) and the integral of 1 is \(x\). Therefore the integral \( \int (x + 1) \, dx \) equals \( \frac{1}{2}x^2 + x \) . This integral can be found either manually or using the computer algebra system.

Step 4: Add the Constant of Integration

Recall that whenever you integrate, a constant of integration must be included in the result. Thus, the final answer for the integral is \( \frac{1}{2}x^2 + x + C \).