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.
1.  What is the Integration of Log x? 
2.  Integral of Ln x Formula 
3.  Integration of Log x Proof 
4.  Definite Integral of ln x 
5.  FAQs on Integral of Log x 
What is the Integration of Log x?
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_{a}x, 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:
Integral of Ln x Formula
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_{a}x with base a is given by, ∫log_{a}x dx = x(log_{a}x  log_{a}e) + C. So, if we substitute a = e in this formula, we have ∫log_{e}x dx = x(log_{e}x  log_{e}e) + 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, ∫log_{10}x dx = x(log_{10}x  log_{10}e) + C
Integration of Log x Proof
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.
Definite Integral of ln x
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,
\(\begin{align}\int_{0}^{1}\ln x \ dx &=\left [ x \ln x  x + C \right ] _{0}^{1}\\&=1 \ln(1)1+C  0\ln(0)+0C\\&=1\times 01+C0+0C\\&=1\end{align}\)
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(log_{10}x  log_{10}e) + C.
 The integral of ln x can be evaluated using the integration by parts method.
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Integration of Log x Examples

Example 1: What is the integration of log x with base 10?
Solution: We can write log x with base 10 as log_{10}x = (log_{e}x/log_{e}10). Therefore, the integral of log x with base 10 is given by,
∫log_{10}x dx = ∫(log_{e}x/log_{e}10) dx
= (1/log_{e}10) ∫log x dx
= (1/log_{e}10)[xlog x  x + C]  [Using the formula for the integration of log x base e]
= xlog x/log_{e}10  x/log_{e}10 + K, where K = C/log_{e}10
= x log_{10}x  x / log_{10}e + K
Answer: ∫log_{10}x dx = x log_{10}x  x / log_{10}e + K, where K is the constant of integration.

Example 2: Find the formula of integral of ln x whole square using the formula of integral of ln x.
Solution: To find the integral of (ln x)^{2}, we will use the integration by parts method whose formula is ∫udv = uv  ∫vdu. Assume u = (ln x)^{2} , dv = dx, then du = (2/x) ln x dx and v = x. Therefore, we have
∫(ln x)^{2} dx = ∫udv
= uv  ∫vdu
= x(ln x)^{2}  ∫x(2/x) ln x dx
= x (ln x)^{2}  2 ∫ln x dx
= x (ln x)^{2}  2 [xln x  x + C]  [Because the integral of ln x is equal to xln x  x + C]
= x (ln x)^{2}  2xln x + 2x  2C
= x [(ln x)^{2}  2ln x + 2] + K, where K = 2C is the integration constant
Answer: ∫(ln x)^{2} dx = x [(ln x)^{2}  2ln x + 2] + K
FAQs on Integration of Log x
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(log_{10}x  log_{10}e) + C. This can be determined using the formula ∫log_{a}x dx = x(log_{a}x  log_{a}e) + 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_{a}x dx = x(log_{a}x  log_{a}e) + C, where C is the integration constant.
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