Question

Evaluate the partial integral. $$ \int_{1}^{e} \frac{y \ln x}{x} d x $$

Short answer

\(\frac{1}{2}y\)

Step-by-step solution

Finding the Antiderivative

To evaluate the definite integral, we first need to find the antiderivative of the function \(f(x) = \frac{y \ln x}{x}\) with respect to \(x\). Using the power rule and the chain rule, we find that the antiderivative of \(f(x)\) is \(F(x) = \frac{1}{2}y \ln^2 x\).

Evaluating the Integral Using the Fundamental Theorem of Calculus

The fundamental theorem of calculus states that the value of a definite integral of a function from a to b is given by the antiderivative of the function evaluated at b minus the antiderivative of the function evaluated at a. For our problem, this would mean evaluating the expression \( F(b)-F(a)\) with \(a =1\) and \(b = e\), giving \( F(e) - F(1) = \frac{1}{2}y \ln^2 e - \frac{1}{2}y \ln^2 1\). Using the property of the natural logarithm that \( \ln e = 1 \) and \( \ln 1 = 0 \), this simplifies to \( \frac{1}{2}y - 0 = \frac{1}{2}y \).

Key concepts

Antiderivative

Understanding the concept of an antiderivative is essential in calculus. It refers to the reverse process of differentiation, and finding an antiderivative means determining a function whose derivative is the given function. For instance, if we need to find an antiderivative of the function \(f(x) = \frac{y \text{ln} x}{x}\), we are looking for a function \(F(x)\) such that \(F'(x) = f(x)\).

This process often involves techniques like the power rule, which allows integration of powers of x, and the chain rule, for composite functions. In the example from the exercise, by applying integration techniques specifically suitable for functions involving natural logarithms, the antiderivative is determined as \(F(x) = \frac{1}{2}y \text{ln}^2 x\).

To effectively improve understanding of antiderivatives:

• Practice finding antiderivatives for different types of functions, such as polynomial, trigonometric, and exponential functions.
• Understand and apply integration rules, such as substitution and integration by parts.
• Use mnemonic devices to remember the forms of common antiderivatives.

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus is a bridge between the concepts of differentiation and integration. It establishes the relationship between the antiderivative and the definite integral of a function. This theorem can be separated into two parts. The first part provides an easy way to find definite integrals when we know an antiderivative of the function. The second part states that the derivative of an integral function is the original function.

In the context of the exercise, the theorem guides us to evaluate the definite integral by taking the antiderivative we found earlier, \(F(x) = \frac{1}{2}y \text{ln}^2 x\), and applying it as \( F(b) - F(a)\), where \(a\) and \(b\) are the limits of integration. This is essentially 'plugging in' these values to the antiderivative to find the area under the curve from \(a\) to \(b\).

For an in-depth understanding of the Fundamental Theorem of Calculus:

• Apply the theorem to various examples to see it in action.
• Recognize how it connects the processes of finding areas under curves (integration) with slopes of tangent lines (differentiation).
• Relate the Fundamental Theorem to real-life scenarios, such as calculating accumulated distances based on speed over time.

Natural Logarithm

The natural logarithm, denoted as \(\text{ln}\), is a logarithm with base \(e\), where \(e\) is an irrational and transcendental number approximately equal to 2.71828. The natural logarithm has profound implications in various disciplines, including calculus, due to its unique properties. One of its key properties is that \(\text{ln} e = 1\) and \(\text{ln} 1 = 0\).

In our example, these properties simplify the process of evaluating the integral. After applying the antiderivative, we utilize the fact that \(\text{ln} e = 1\) and \(\text{ln} 1 = 0\) to find that \(F(e)\) becomes simply \(\frac{1}{2}y\), and \(F(1)\) is 0, yielding the final answer of \(\frac{1}{2}y\) as the integral's value.

To grasp the natural logarithm better:

• Practice simplifying expressions involving the natural logarithm.
• Explore its relationship with the exponential function \(e^x\).
• Understand how its derivative, \(\frac{1}{x}\), makes it a fundamental tool in integration techniques like logarithmic differentiation.