Question

Evaluate the definite integral. $$ \int_{1}^{e} 2 x \ln x d x $$

Short answer

The definite integral of \( \int_{1}^{e} 2 x \ln x dx \) is \( e^2 - e^2/2 + 1/2 \).

Step-by-step solution

Identifying u and dv

Identify the parts of the integrand that will be \( u \) and \( dv \). This is important because the integral is of the form \( u dv \). Here, \( u = \ln x \) and \( dv = 2x dx \).

Compute du and v

Compute the differential of \( u \) and the integral of \( dv \) to get \( du \) and \( v \). The derivative of \( u = \ln x \) is \( du = 1/x dx \). The integral of \( dv = 2x dx \) is \( v = x^2 \). Now we have all parts to use for the formula.

Apply the Integration by Parts Formula

Apply the integration by parts formula: \( \int u d v = u v - \int v d u \). Plugging in our parts, this gives us: \( \int_{1}^{e} \ln x \cdot 2x dx = [ x^2 \ln x]_{1}^{e} - \int_{1}^{e} x^2 \cdot (1/x) dx = [ x^2 \ln x]_{1}^{e} - \int_{1}^{e} x dx \).

Evaluate the New Integral

The integral we have now is easier, so evaluate \( \int_{1}^{e} x dx = [x^2/2]_{1}^{e} = e^2/2 - 1/2 \).

Evaluate the Definite Integral

The final step is to evaluate the definite integral with original bounds: \( [ x^2 \ln x]_{1}^{e} - (e^2/2 - 1/2) = e^2 - e^2/2 + 1/2 \).

Key concepts

Definite Integral

Understanding the definite integral is crucial for solving many problems in calculus. A definite integral represents the accumulation of quantities, such as area under a curve, over a specific interval. In mathematical terms, you can write it as \[ \int_{a}^{b} f(x) dx \], where \( f(x) \) is a function and \( [a, b] \) are the lower and upper limits of integration, respectively.

When evaluating the definite integral, you calculate the difference between the values of an antiderivative of the function at \( b \) and \( a \)—effectively finding the total 'change' across the interval. This gives the net value after considering the entire range of the variable, a concept that comes in handy when calculating physical quantities like work done or the distance travelled.

Natural Logarithm

The natural logarithm, denoted as \( \ln(x) \) and based on the constant \( e \approx 2.71828 \), is the amount of time needed to reach a certain level of continuous growth. It's an essential function in calculus because of its relationship to exponential functions. In integration and differentiation, the natural logarithm interacts elegantly with these functions, often simplifying otherwise complex expressions.

The derivative of \( \ln(x) \) is \( \frac{1}{x} \) and arises naturally when applying integration techniques such as integration by parts, especially when the function involves \( \ln(x) \) itself. This property is very useful for integrating products of polynomials and logarithmic functions.

Integration Techniques

There are various techniques to tackle integrals, one being integration by parts—a method based on the product rule for differentiation. This technique is especially helpful when integrating the product of two functions that are not easily integrable in their original form. The basic formula for integration by parts is \[ \int u dv = uv - \int v du \].

To apply this, one must wisely choose \( u \) and \( dv \) from the given integral. Afterwards, find \( du \) (the derivative of \( u \) with a slight rearrangement) and \( v \) (the antiderivative of \( dv \) ). Following these steps transforms the original integral into a form that can be simplified and solved more straightforwardly.

Calculus

Calculus, with its two major branches—differential and integral calculus—is the mathematical study of change and accumulation. Differential calculus focuses on finding rates of change, such as velocity, and slopes of curves, while integral calculus is about accumulation of quantities and areas under curves.

The concepts of integral calculus, including the definite integral and various techniques like integration by parts, are fundamental for solving problems in physics, engineering, economics, and beyond. Mastering these concepts not only allows for the resolution of mathematical problems but also provides a powerful tool for analyzing and understanding a multitude of real-world situations.