Question

Logarithm base \(b\) Prove that \(\int \log _{b} x d x=\frac{1}{\ln b}(x \ln x-x)+C\).

Short answer

Question: Find the integral of the function \(\log_b x\). Answer: The integral of the function \(\log_b x\) is given by \(\int \log _{b} x d x = \frac{1}{\ln b}(x \ln x - x) + C\).

Step-by-step solution

Change the base of logarithm from \(b\) to \(e\) using logarithmic identities

First, we need to rewrite \(\log_b x\) in terms of natural logarithm \(\ln x\). According to the logarithmic identity, \(\log_b x = \frac{\ln x}{\ln b}\). Now the integral becomes: $$\int \log _{b} x d x = \int\frac{\ln x}{\ln b}dx$$

Perform Integration by Parts

To solve the integral, we will use Integration by Parts. According to Integration by Parts, if we have an integral of the form \(\int udv\), it can be written as \(uv - \int vdu\). Let \(u= \ln x\) and \(dv= \frac{1}{\ln b} dx\). We need to differentiate \(u\) and integrate \(dv\): $$du = \frac{1}{x} dx$$ $$v = \int\frac{1}{\ln b} dx = \frac{x}{\ln b}$$ Now, we can write the integral using the Integration by Parts formula: $$\int\frac{\ln x}{\ln b}dx = \frac{1}{\ln b}(x\ln x) - \int \frac{1}{\ln b} \cdot \frac{x}{x} dx$$

Simplify the integral and integrate

Simplify the integral to make it easier to solve: $$\int\frac{\ln x}{\ln b}dx = \frac{1}{\ln b}(x\ln x) - \frac{1}{\ln b} \int dx$$ Now, integrate the remaining integral: $$\int\frac{\ln x}{\ln b}dx = \frac{1}{\ln b}(x\ln x) - \frac{1}{\ln b} (x + C_1)$$

Combine constants

Finally, combine the constants by multiplying the constant term by \(\frac{1}{\ln b}\): $$\int\frac{\ln x}{\ln b}dx = \frac{1}{\ln b}(x\ln x) - \frac{1}{\ln b}(x) + \frac{C_1}{\ln b}$$ Now, denote \(\frac{C_1}{\ln b}\) as a new constant \(C\): $$\int\frac{\ln x}{\ln b}dx = \frac{1}{\ln b}(x\ln x - x) + C$$ Thus, it's proved that $$\int \log _{b} x d x = \frac{1}{\ln b}(x \ln x - x) + C$$

Key concepts

Logarithmic Integration

Logarithmic integration involves integrating functions with logarithms in them. For example, consider the integration of a logarithmic function with base 'b'. First, it's essential to express the logarithm with base 'b' as a natural logarithm, taking advantage of the identity \( \log_b x = \frac{\ln x}{\ln b} \) to make the function easier to integrate.

Once the base is changed, typical integration methods, such as integration by parts, can be used to find the solution. Logarithmic integration is a handy technique when dealing with growth and decay problems or when determining the area under a curve represented by a logarithmic function.

Natural Logarithm

The natural logarithm, denoted \( \ln \) , refers to the logarithm to the base \( e \) , where \( e \) is an irrational and transcendental number approximately equal to 2.71828. It is widely used in calculus because of its simple derivative, \( \frac{d}{dx}\ln x = \frac{1}{x} \) , and because it serves as the inverse function to the natural exponential function \( e^x \) .

The natural logarithm also arises naturally in many areas, such as compound interest, Euler's identity, and population growth models. Its properties make it essential for solving equations involving exponential growth and decay.

Integration Techniques

Integration by Parts

Integration techniques are methods used to find the antiderivatives of functions. One of the advanced techniques for integration is integration by parts, which is based on the product rule for differentiation. The formula for integration by parts is \( \int u dv = uv - \int v du \) where \( u \) and \( dv \) are chosen from the original integral. The choice of \( u \) and \( dv \) is crucial and can make the difference between an easy solution and a complicated or even unsolvable integral. Typically, \( u \) is chosen as the part of the integral that becomes simpler when differentiated, while \( dv \) is chosen as the part that can easily be integrated.

Calculus

Calculus is a branch of mathematics focused on rates of change (differential calculus) and accumulation of quantities (integral calculus). It is used to solve problems in fields such as physics, engineering, economics, statistics, and medicine.

One major application of calculus is in the analysis of motion and change, where derivatives measure rates of change and integrals represent the total accumulation of quantities. Calculus also provides tools to work with complex functions that describe real-world phenomena, enabling precise predictions and deep understanding of nature's behaviors.