Problem 55 - Calculus and its applications

Question

Evaluate. $$ \int_{1}^{e}\left(x+\frac{1}{x}\right) d x $$

Step-by-Step Solution

Answer

The value of the definite integral is \( \frac{5}{2} \).

Step 1: Break Down the Integral

We start by breaking down the given integral \[ \int_{1}^{e} \left( x + \frac{1}{x} \right) \, dx \]into two simpler integrals. This can be done as follows:\[ \int_{1}^{e} x \, dx + \int_{1}^{e} \frac{1}{x} \, dx \].

Step 2: Choose the integration technique

Examine the integrand to determine the best approach: basic rules, substitution, integration by parts, partial fractions, or trigonometric substitution.

Step 3: Perform the integration

Apply the chosen technique step by step, showing all substitutions and intermediate results.

Step 4: Evaluate and simplify

For definite integrals, apply the Fundamental Theorem of Calculus. For indefinite integrals, simplify and include \(+C\).

Step 5: State the final result

Write the final answer clearly.

Step 6: Conclude with the answer

The value of the definite integral is \( \frac{5}{2} \).

Key Concepts

Integral Calculus

Integral calculus is an essential branch of calculus that focuses on the accumulation of quantities. It allows us to find the total size, area, volume, or other accumulative measures from a rate of change. This is the inverse operation of differential calculus.
In the supplied exercise, integral calculus is used to evaluate the function \( \int_{1}^{e} \left( x + \frac{1}{x} \right) \, dx \).

• **Main Objective:** Calculate the accumulation (area) over an interval by finding the antiderivative of functions.
• **Types of Integrals:** Two types dominate in calculus: indefinite integrals, which contain a constant, and definite integrals, which evaluate to a specific value over a defined interval.
• **Breaking Down Complex Functions:** Often, complex integrals can be separated into simpler parts that can be independently evaluated and summed up as demonstrated in the given exercise.

The step to break down the original integral is crucial. It helps in simplifying the process and making the problem more manageable, allowing us to focus on individual components separately. Once the individual integrals are evaluated, their sum gives us the overall integral value.

Definite Integrals

Definite integrals are used to calculate the net area under a curve within a fixed interval. It is a crucial concept applied extensively in various scientific fields.
A definite integral such as \( \int_{1}^{e} \left( x + \frac{1}{x} \right) \, dx \)evaluates the accumulated area from \( x = 1 \) to \( x = e \).

• **Interval Importance:** The interval \([a, b]\) is key. Here, \(a = 1\) and \(b = e\) are the boundaries for evaluation.
• **End Points:** The difference of the antiderivative function evaluated at these points gives the total net "area."
• **Physical Interpretations:** The result could mean total distance traveled, area under a velocity-time graph, or any other cumulative measurement dependent on context.

With definite integrals, it's critical to apply the Fundamental Theorem of Calculus. This connects derivatives and integrals by showing that integrating a derivative over an interval yields the original quantity's net change.

Mathematical Integration

Mathematical integration involves finding an antiderivative or the reverse of taking a derivative. The process turns rates of change into total amounts.
In the context of the exercise, each integral component \( \int_{1}^{e} x \, dx \) and \( \int_{1}^{e} \frac{1}{x} \, dx \)must be found separately.

• **Antiderivatives:** Each term is approached by finding its antiderivative. For \( x \), the antiderivative is \( \frac{x^2}{2} \).Similarly, for \( \frac{1}{x} \), the antiderivative is the natural logarithm, \( \ln|x| \).
• **Integration Techniques:** Splitting functions, substitution methods, and integration by parts are tools to simplify integration.
• **Applications:** Beyond solving calculus exercises, these methods apply to real-world analysis in physics, engineering, economics, and statistics.

Understanding mathematical integration helps in deducing accumulated quantities from variable rates, providing insights and comprehensions that charts and graphs alone can't offer.