Question

The value of \((e)^{2} \int_{e}[d x /(\log x)]-2 \int_{1}\left(e^{x} / x\right) d x\) is...... (a) \(\mathrm{e}^{2}\) (b) e (c) \((1 / e)\) (d) 0

Short answer

The value of the given expression is not a real number; the correct answer is not among the given options. There may be a typo in the original problem statement, leading to the incomplete integral for the natural logarithm, which causes the expression to be infinite. Please recheck the problem statement and verify that the exercise is provided correctly.

Step-by-step solution

Evaluate the first integral

Begin with the first integral, \(\int_{e}[d x /(\log x)]\). We can rewrite this integral as \(\int_{e}[\frac{1}{\ln{x}}dx]\). Now, observe that its antiderivative function is given by \(u=\ln{x}\) because of definition of natural logarithm \(\ln e^u = u\). Firstly, find the derivative of \(u\) with respect to \(x\): \[du = \frac{1}{x} dx\] Now, make the substitution \(u = \ln{x}\) into the integral: \[\int_{e}[\frac{1}{\ln{x}}dx] = \int_{1}^{\infty}du = u \Big|_{1}^{\infty} = (\infty - 1)\]

Evaluate the second integral

Now, move onto the second integral: \(-2 \int_{1}\left(e^{x} / x\right) d x\). The antiderivative of this second integral is not elementary, so we can't find an explicit expression for it. However, we can still calculate the integral using properties of natural logarithmic function. We know that integrals are linear operators, meaning that the integral of a sum of functions is equal to the sum of the integral of each function. We can apply this property in the form of integration by parts: Let \(u= x\) and \(dv = e^x dx\). Then, \(du = dx\) and \(v = e^x\). Using integration by parts: \[-2 \int_{1}\left(e^{x} / x\right) d x = - 2 \Big[u v - \int v du\Big]_1 = -2\left[x e^x - \int_{1} e^xdx\right]_1\] Now, integrate \(e^x\) with respect to \(x\): \[-2\left[x e^x - e^x\right]_1 = -2 \left[(e^x - xe^x)\Big|_{1} - (e^1 - (1)e^1)\right]\]

Combine the results and determine the correct option

Now, we will combine the results from Step 1 and Step 2: \((e)^{2} \int_{e}[d x /(\log x)]-2 \int_{1}\left(e^{x} / x\right) d x = e^2(\infty - 1) + -2 \left[(e^x - xe^x)\Big|_{1} - (e^1 - (1)e^1)\right]\) Since \((\infty - 1)\) is still infinite, the value of the overall expression is not a real number; the correct answer is not among the given options. However, there may be a typo in the original problem statement, leading to the incomplete integral for the natural logarithm, which causes the expression to be infinite. If that is the case, please recheck the problem statement and verify that the exercise is provided correctly.

Key concepts

Definite Integrals

Definite integrals are a fundamental concept in calculus, providing a method to calculate the area under a curve within a given interval. Specifically, the definite integral from a to b of function f is denoted as \(\int_{a}^{b} f(x) dx\) and is calculated using the Fundamental Theorem of Calculus. This process involves finding an antiderivative of the function, which is a function F such that F'(x) = f(x), and then evaluating F(b) - F(a).

It is important to note that the definite integral has both upper and lower limits, therefore it represents the net area between the function and the x-axis on the interval [a, b]. If the function passes below the x-axis, the area is considered negative, reflecting the directionality of integration. In problems involving natural logarithms, like the one given, it is essential to ensure that integrals are properly set up, as logarithmic functions approach negative infinity near zero and infinity at large positive values, which can affect the existence of the definite integral.

Natural Logarithm Properties

The natural logarithm, denoted as ln(x), is the inverse function of the exponential function e^x. It has unique properties that are particularly useful when dealing with integrals. For example, ln(e) = 1, because e is the base of natural logarithms. Additionally, ln(x) can be used to transform products into sums, using the property ln(ab) = ln(a) + ln(b), and powers into products, with ln(a^b) = b * ln(a). This can simplify the integration of functions involving e^x or ln(x).

The logarithm properties also play a critical role in integration by parts, where the choice of the functions u and dv often involves the natural logarithm due to its derivative being \(1/x\). It is also worth noting that the definite integral containing the natural logarithm must respect its domain, which only includes positive real numbers, to avoid undefined expressions.

Linear Operators in Integration

Integrals, much like derivatives, are linear operators. This means they have two key properties: additivity and homogeneity. Additivity allows us to integrate a sum of functions separately as \(\int (f + g)dx = \int fdx + \int gdx\). Homogeneity lets us pull out a constant from an integral, so \(\int cf(x)dx = c\int f(x)dx\), where c is any constant. These properties are instrumental when solving complex integrals as they allow us to break them down into more manageable pieces.

In the case of integration by parts, a technique that derives from the product rule in differentiation, we make strategic choices for u and dv to simplify the integral. For instance, in our exercise, the function e^x/x could be decomposed, and by applying these linear operator properties, the integration becomes more tractable, albeit with careful consideration of the appropriate function selection for u and dv to ensure the resulting integrals are calculable.