Integration of Log x

The integration of log x is equal to xlogx - x + C, where C is the integration constant. We can evaluate the integral of ln x (integration of log x with base e) using the integration by parts formula (also known as the UV formula of integration). The integral of a function gives the area under the curve of the function. Therefore, the integral of ln x gives the area under the curve of the function f(x) = ln x. Mathematically, we can write the formula for the integration of log x, ∫log x dx = xlogx - x + C (OR) ∫ln x dx = xlnx - x + C, where log x or ln x are the natural logarithmic function.

Further in this article, we will evaluate the integral of ln x or log x with base e using the integration by parts formula. In this article, we are considering log x with base e by default. We will also calculate the definite integration of log x with different limits and solve examples using the integral of ln x for a better understanding of the concept.

The integration of log x with base e is equal to xlogx - x + C, where C is the constant integration. The logarithmic function is the inverse of the exponential function. Generally, we write the logarithmic function as log_ax, where a is the base and x is the index. The integral of ln x can be calculated using the integration by parts formula given by ∫udv = uv - ∫vdu. The integration of log x gives the area under the curve f(x) = log x. Mathematically, we write the integration of log x as ∫log x dx = xlogx - x + C, where ∫ is the integration symbol, dx shows the integral of ln x is with respect to x and C is the integration constant. Let us now explore its formula in the next section:

The formula for the integration of ln x dx is given by, ∫ln x dx = xlnx - x + C. We can also write the formula as ∫log x dx = xlogx - x + C, where we are considering logarithmic function log x with base e. The integral of logarithmic function log_ax with base a is given by, ∫log_ax dx = x(log_ax - log_ae) + C. So, if we substitute a = e in this formula, we have ∫log_ex dx = x(log_ex - log_ee) + C = ∫ln x dx = xlnx - x + C [Because logarithmic function with base e is generally written as ln x]. Th image below shows the formula integral of ln x:

Please note that if the base of the logarithmic function is 10, then its integration is given by, ∫log₁₀x dx = x(log₁₀x - log₁₀e) + C

Now that we know that the formula for the integration of log x with base e is equal to ∫log x dx = xlogx - x + C, we will prove this formula using integration by parts. The formula for the integration by parts method of integration is ∫udv = uv - ∫vdu, where u and v are functions. We can write log x as log x × 1. We will choose the functions u and v according to the sequence given by ILATE (Inverse Trigonometric Function, Logarithmic Function, Algebraic Function, Trigonometric Function, Exponential Function). Assume u = log x and dv = dx. This implies du = 1/x dx and v = x [Because the derivative of log x with base e is equal to 1/x]. Therefore, we have

∫(log x × 1) dx = x × log x - ∫x (1/x) dx

= xlog x - ∫dx

= xlog x - x + C

Hence, we have proved that the integration of log x is equal to x log x - x + C, where log x has base e.

The formula for the integral of ln x is given by, ∫ln x dx = xlnx - x + C, where C is the constant of integration. In this section, we will calculate the definite integration of log x with different limits.

Integral of Ln x From 0 to 1

The integral of ln x with limits from 0 to 1 is given by,

\(\begin{align}\int_{0}^{1}\ln x \ dx &=\left [ x \ln x - x + C \right ] _{0}^{1}\\&=1 \ln(1)-1+C - 0\ln(0)+0-C\\&=1\times 0-1+C-0+0-C\\&=-1\end{align}\)

Hence, the value of the integral of ln x from 0 to 1 is equal to -1.

Important Notes on Integral of Log x

• The integration of log x with base e is equal to xlog x - x + C, where C is the constant of integration.
• The integration of log x with base 10 is equal to x(log₁₀x - log₁₀e) + C.
• The integral of ln x can be evaluated using the integration by parts method.

☛ Related Topics:

• Integral of Cot2x
• Integral of 2x
• Integral of 2sinx

What is Integration of Log x in Calculus?

The integration of log x is equal to xlogx - x + C, where C is the integration constant. Here, the function log x is considered with base e which is an exponential number.

What is the Integration of Log x Formula?

The formula for the integration of ln x dx is given by, ∫ln x dx = xlnx - x + C. We can evaluate the integral of ln x (integration of log x with base e) using the integration by parts formula

How to Find the Integration of Ln x dx?

The integral of ln x can be calculated using the integration by parts formula given by ∫udv = uv - ∫vdu. In this formula, we assume u = ln x and dv = dx and solve the integral using the formula.

What is the Integration of Log x With Base 10?

The integration of log x with base 10 is equal to x(log₁₀x - log₁₀e) + C. This can be determined using the formula ∫log_ax dx = x(log_ax - log_ae) + C, where we substitute a = 10.

What is the Integration of Log x Whole Square?

The integral of log x whole square is equal to ∫(log x)² dx = x [(log x)² - 2log x + 2] + K.

What is the Formula of Integral of Log x with Base a?

The formula of integral of log x with base a is given by ∫log_ax dx = x(log_ax - log_ae) + C, where C is the integration constant.