Problem 20 - AP Calculus with 8 Practice Tests 15th

Question

\(\int \frac{x^{3}-x-1}{x^{2}} d x=\) (A) \(\frac{\frac{1}{4} x^{4}-\frac{1}{2} x^{2}-x}{\frac{1}{3} x^{3}}+C\) (B) \(1+\frac{1}{x^{2}}+\frac{2}{x^{3}}+C\) (C) \(\frac{x^{2}}{2}-\ln |x|-\frac{1}{x}+C\) (D) \(\frac{x^{2}}{2}-\ln |x|+\frac{1}{x}+C\)

Step-by-Step Solution

Answer

The correct answer is (D).

Step 1: Simplify the Integrand

Given the integral \( \int \frac{x^{3} - x - 1}{x^{2}} \, dx \), simplify the integrand by dividing each term in the numerator by \( x^2 \). This results in \( \int (x - \frac{1}{x} - \frac{1}{x^2}) \, dx \).

Step 2: Split the Integral

Now, split the integral into three separate integrals: \( \int x \, dx - \int \frac{1}{x} \, dx - \int \frac{1}{x^2} \, dx \).

Step 3: Integrate Each Term

- Integrate \( x \) to get \( \frac{x^2}{2} \).- Integrate \( \frac{1}{x} \) to get \( \ln |x| \).- Integrate \( \frac{1}{x^2} \) (or \( x^{-2} \)) to get \( -\frac{1}{x} \).

Step 4: Combine the Results

Combine the results from Step 3 to get the full integral: \( \frac{x^2}{2} - \ln |x| + \frac{1}{x} + C \).

Step 5: Compare with Given Options

Review the provided options:- (A) \( \frac{\frac{1}{4} x^{4}-\frac{1}{2} x^{2}-x}{\frac{1}{3} x^{3}}+C \)- (B) \( 1+\frac{1}{x^{2}}+\frac{2}{x^{3}}+C \)- (C) \( \frac{x^{2}}{2}-\ln |x| - \frac{1}{x}+C \)- (D) \( \frac{x^{2}}{2}-\ln |x|+\frac{1}{x}+C \)Identify that option (D) matches the combined result formula.

Key Concepts

Integration Techniques

Integration techniques are essential in learning how to find the integral of various functions. They include methods like substitution, integration by parts, and partial fraction decomposition. In this exercise, the technique primarily utilized is polynomial long division and separation of terms.

Polynomial long division is crucial when you encounter rational functions, just like in this exercise, where you divide the numerator by the denominator: \( \frac{x^3 - x - 1}{x^2} \). By breaking down this expression, it becomes easier to handle: \( x - \frac{1}{x} - \frac{1}{x^2} \).

Separation of terms means breaking down a complex integrand into simpler parts. Each of these can be integrated individually. So when you separate, \( \int x \, dx - \int \frac{1}{x} \, dx - \int \frac{1}{x^2} \, dx \) emerges. This step simplifies the integration process as you apply the rules individually to each term.

Definite Integrals

Definite integrals are used to calculate the area under a curve between two points. Unlike indefinite integrals, they provide a numerical value.

When working with definite integrals, you evaluate the integral first and then apply the limits of integration. Here, the exercise focuses on indefinite integration, but understanding definite integrals helps when solving related problems.

For instance, if our integrand were \( \int_{a}^{b} (x - \frac{1}{x} - \frac{1}{x^2}) \, dx \), after obtaining the integral \( \frac{x^2}{2} - \ln |x| + \frac{1}{x} \), you would substitute the limits \( a \) and \( b \) into this expression and find the difference to get the area.

Indefinite Integrals

Indefinite integrals represent a family of functions and include a constant of integration, usually denoted as \( C \). This constant arises because integration is the reverse process of differentiation, and differentiation removes constant terms.

In this exercise, after integrating the separated terms \( \int x \, dx \), \( \int -\frac{1}{x} \, dx \), and \( \int -\frac{1}{x^2} \, dx \), you obtain the result: \( \frac{x^2}{2} - \ln |x| + \frac{1}{x} + C \). Each term represents the antiderivative of the original expression.

Remember, the presence of \( C \) signifies potential vertical shifts in the function's graph, indicating that there are infinitely many solutions differing by a constant. Indefinite integrals are crucial for solving differential equations and analyzing function behavior.