Problem 59 - AP Calculus with 8 Practice Tests 15th

Question

\(\int \sqrt{x}(\sqrt{x}-1) d x=\) (A) \(2\left(x^{3 / 2}-x\right)+C\) (B) \(\frac{x^{2}}{2}-x+C\) (C) \(\frac{1}{2}(\sqrt{x}-1)^{2}+C\) (D) \(\frac{1}{2} x^{2}-\frac{2}{3} x^{3 / 2}+C\)

Step-by-Step Solution

Answer

The correct solution is (D) \(\frac{1}{2} x^{2}-\frac{2}{3} x^{3 / 2}+C\).

Step 1: Expand the integrand

First, we need to simplify the expression \(\sqrt{x}(\sqrt{x} - 1)\). Distribute \(\sqrt{x}\) to both terms inside the parentheses:\[\sqrt{x}(\sqrt{x} - 1) = (\sqrt{x})(\sqrt{x}) - (\sqrt{x})(1) = x - \sqrt{x}.\]

Step 2: Rewrite and Separate the Integral

Now express the integral as two separate integrals based on the simplified expression:\[\int x \; dx - \int \sqrt{x} \; dx.\]

Step 3: Integrate Each Term Separately

Calculate each integral separately.For \(\int x \, dx\), the antiderivative is:\[\frac{x^2}{2} + C_1\]For \(\int \sqrt{x} \, dx\), write \(\sqrt{x}\) as \(x^{1/2}\) and use the power rule:\[\int x^{1/2} \, dx = \frac{x^{1/2 + 1}}{1/2 + 1} = \frac{x^{3/2}}{3/2} = \frac{2}{3}x^{3/2} + C_2\]

Step 4: Combine the Results

Subtract the results from Step 3:\[\frac{x^2}{2} - \frac{2}{3}x^{3/2} + C.\]Combine the constants \(C_1\) and \(C_2\) into a single constant \(C\).

Step 5: Match the Expression with Options

Compare the final result to the given options:The expression we found is \(\frac{x^2}{2} - \frac{2}{3} x^{3/2} + C\), which matches with option D.

Key Concepts

Calculus

Calculus is a branch of mathematics focused on change and motion. It has two main branches: differential calculus and integral calculus. Differential calculus studies how things change, focusing on concepts like the derivative. Integral calculus, on the other hand, is centered around accumulation and area under curves. The definite integral, which is part of integral calculus, allows us to calculate the accumulation of quantities, such as areas or total change over an interval.
Understanding calculus opens up the ability to solve real-world problems in fields like physics, engineering, and economics. It is fundamental for describing and predicting systems where movement or change occurs.

Antiderivative

An antiderivative is a function whose derivative is equal to the original function. It is the reverse process of differentiation and is a key concept in finding integrals. For instance, if the derivative of a function, say \( f(x) \), is given as \( f'(x) = g(x) \), then \( f(x) \) is the antiderivative of \( g(x) \).

• Antiderivatives are not unique. They often have a constant \( C \) added, since the derivative of a constant is zero.
• Find antiderivatives to compute indefinite integrals, representing a family of functions.

Grasping the concept of antiderivatives is crucial, as they provide the foundation for integration techniques.

Integration Techniques

Integration techniques are methods used to find the integral or antiderivative of a function. Various techniques exist, each suited to different types of functions and problems:

• Substitution Method: When there's a composition of functions, substituting can simplify the integral.
• Integration by Parts: Useful when the integrand is the product of two functions.
• Partial Fraction Decomposition: Applicable for rational functions, breaking them into simpler fractions.
• Trigonometric Integrals: Deals with integrals involving trigonometric functions.

Each technique aims to simplify the integration process, enabling us to solve complex problems. In this exercise, the expanded expression \( x - \sqrt{x} \) was integrated using basic techniques that rely on transforming the initial problem into simpler parts.

Power Rule for Integration

The Power Rule for Integration is a basic and essential technique for finding antiderivatives of polynomial expressions. If you're familiar with differentiation, this will sound quite similar to the power rule there. The rule is applicable when integrating functions of the form \( x^n \), where \( n \) is any real number not equal to -1.Here is how it works: - For a function \( x^n \), the integral is \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \).- The constant \( C \) is added because integration results in a family of functions.- Remember, the power rule does not apply when \( n = -1 \) (use the natural logarithm rule for \( \frac{1}{x} \)).
In the given exercise, the power rule was used to find the antiderivative of \( x \) and \( x^{1/2} \), simplifying the integration process and ultimately leading to the correct solution.