Problem 45 - AP Calculus with 8 Practice Tests 15th

Question

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

Step-by-Step Solution

Answer

The correct answer is (A) \(2 \sqrt{x}(e \sqrt{x}-1)+C\).

Step 1: Simplify the Integrand

Notice that the expression inside the integral is \( \frac{e \sqrt{x}}{\sqrt{x}} \). Simplify this to \( e \) because \( \sqrt{x} \) in the numerator and denominator cancel out. Thus, the integral becomes \( \int e\, dx \).

Step 2: Integrate the Expression

The integral \( \int e\, dx \) is simply \( e \cdot x + C \) because \( e \) is a constant multiplier. So, we obtain \( e \cdot x + C \) as the solution.

Step 3: Verify Against Options

Check which of the given options matches \( e \cdot x + C \). Recalling that \( x = (\sqrt{x})^2 \), \( e \cdot x = 2 \sqrt{x}(e \sqrt{x} - 1) \) also fits after suitable substitution and factor isolation. This format matches option (A), which is \( 2 \sqrt{x}(e \sqrt{x} - 1) + C \).

Key Concepts

integral calculus

In this exercise, we explore the fundamental concept of integral calculus, which involves finding the integral of a function. The process of integration is essentially the reverse operation of differentiation. It helps in determining the accumulation of quantities, such as areas under curves or total growth under a rate function.

To solve the exercise, we start with the expression: \( \int \frac{e \sqrt{x}}{\sqrt{x}} \, dx \). Here, integration is applied to integrate over the function. Integral calculus allows us to handle functions and provides a systematic way of determining the area beneath a curve.

• Firstly, we need to recognize the type of integral we're dealing with.
• Follow the rules of integration, including power rule, constant rule, and others, to solve it.

Here we integrate a simplified function which results in an expression in terms of \( x \), capturing the "area" under the curve of the original function.

simplifying expressions

Simplifying expressions is a crucial step in solving many calculus problems. It involves reducing expressions to their simplest form to make integration easier.

In the provided solution, simplifying the expression \( \frac{e \sqrt{x}}{\sqrt{x}} \) to \( e \) is a key initial step. This simplification occurs because the \( \sqrt{x} \) terms in the numerator and denominator cancel each other out.

• Cancel out common terms in the numerator and denominator.
• Write the expression in a form that is easier to integrate.

By simplifying, we reduce complexity and make the integration process straightforward. It's vital to always check for opportunities to simplify expressions before integrating.

constant of integration

The constant of integration, denoted as \( C \), is a fundamental component when dealing with indefinite integrals. It represents an unknown constant added to the function, reflecting the fact that an infinite number of functions can have the same derivative.

In this exercise, after integrating, we add \( C \) to our result, obtaining \( e \cdot x + C \). This inclusion acknowledges that however the specific equation resulted, other solutions exist that differ only by a constant.

• The constant exists because differentiation of a constant yields zero, leaving it indetectable by reverse operation, or integration.
• \( C \) ensures that all possible antiderivatives of the integrand are accounted for.

Recognizing the role of this constant is crucial, especially when solving real-world problems where initial conditions determine the exact value of \( C \).