Introduction to the integral of ln x

In calculus, the integral is a fundamental concept that assigns numbers to functions to define displacement, area, volume, and all those functions that contain a combination of tiny elements. It is categorized into two parts, definite integral calculator and indefinite integral. The process of integration calculates the integrals. This process is defined as finding an antiderivative of a function. 

Integrals can handle almost all functions, such as trigonometric, algebraic, exponential, logarithmic, etc. This article will teach you what is integral to a natural log function ln x. You will also understand how to compute integration of lnx by using different integration techniques.

What is the integral of lnx?

The integration of ln x is an antiderivative of the ln x function which is equal to xln x-x. It is also known as the reverse derivative of ln x function which is a trigonometric identity. 

The function ln x is a logarithm to the base e, where e is the Euler’s number equals to 2.7182. The function ln x is also known as the natural log of x. 

Integral of ln(x) formula

The formula of integral of ln contains integral sign, coefficient of integration and the function as ln(x). It is denoted by ∫(ln x)dx. In mathematical form, the integral of ln x is:

 $∫\ln xdx = x\ln x - x + c$

 Where c is any constant involved, dx is the coefficient of integration and ∫ is the symbol of integral. 

How to calculate the integral of ln(x)?

The antiderivative of lnx is its antiderivative that can be calculated by using different integration techniques. In this article, we will discuss how to calculate integral of ln x by using following integral techniques:

1. U-substitution method
2. Integration by parts
3. Definite integral

Integral of log x by using u-substitution method

The u substitution is a method of integration in calculus. It is used to calculate the integral of a function which is complex to be calculated by usual integration. To eveluate the antiderivative of ln x, you can use an online tool like u subsitution calculator. Let’s discuss calculating the integration of ln x by using u-substitution.

Proof of Integral of natural log by using substitution method

To proof the integral of ln x by using substitution method, suppose that:

$I=\int \ln xdx$

Suppose that,

$u=\ln x$

and

$du = \frac{1}{x} dx$

Or,

$xdu = dx$

And, 

$x=e^u$

Now substituting these values in the above integral.

$I=\int ue^udu$

Now integrating by using the integration by parts formula,

$I=f(x)\int g(x)dx-\int[\frac{d}{dx}(f(x)\int g(x)dx]dx$

Assuming that, 

$f(u)=u$ and $g(u)=e^u$

Then, by using above formula, 

$I=u\int e^udu-\int[\frac{d}{du}(u)\int e^udu]du$

Since the integral of e^x is e^x,

$I=ue^u-\int e^udu$

$I=ue^u-e^u$

Now substituting the value of u, we get

$I=x\ln x-x+c$

Hence we have verified the antiderivative of lnx by using the u-substitution method.

Integral of ln x by using integration by parts

The derivative of a function calculates the rate of change, and integration is the process of finding the antiderivative of a function. The integration by parts is a method of solving the integral of two functions combined together. The formula of integral by parts calculator is;

$I=f(x)\int g(x)dx-\int \left(\frac{d}{dx}(f(x)\int g(x)dx)\right dx$

Let’s discuss calculating the integral of ln(x) by using integration by parts.

Proof of integration of lnx by using integration by parts

To integrate the function ln(x) using integration by parts, we can use the following formula:

$I=f(x)\int g(x)dx-\int \left(\frac{d}{dx}(f(x)\int g(x)dx)\right dx$

Suppose that, 

$f(x) = \ln x$

$g(x) =1$ 

Using these values in the above formula, 

$I=\ln x\int 1dx-\int (\frac{d}{dx}(x)\int 1dx)dx$

Integrating, 

$I=x\ln x-(\frac{1}{x}\times x)dx$

$I=x\ln x-\int 1dx$

$I=x\ln x-x+c$

Hence the antiderivative of ln x is x ln x - x+c, where c is a constant.

Integral of natural log by using definite integral 

The definite integral is a type of integral that calculates the area of a curve by using infinitesimal area elements between two points. The definite integral can be written as:

$∫^b_a f(x) dx = F(b) – F(a)$

The above formula can also be used to calculate arc length of an arc. You can also use the arc length calculator for this. Let’s understand the verification of the integration of ln x by using the indefinite integral.

Proof of integral of log x by using definite integral

To compute the integral of ln by using a definite integral, we can use the interval from 0 to . Let’s compute the ln integral from 0 to 1. For this we can write the integral as:

$∫^1_0 \ln x dx = |x\ln x - x|^1_0$ 

Now, substituting the limit in the given function.

$∫^1_0 \ln x dx = [(1)\ln (1) - 1]-[(0)\ln 0 -0]$

$∫^1_0 \ln x dx = \ln(1)-1$

Since ln 1 is equal to 0, therefore, 

$∫^1_0 \ln x dx = -1$ 

Which is the calculation of the definite integral of log x. 

Frequently Asked Questions

What is the antiderivative of ln x dx?

The antiderivative or the integral of the natural log is equal to the difference of xln x and x. It can be obtained by using difference integration techniques such as the u-substitution, integration by parts and the definite integral of ln x.

What is the relation between ln and log?

The natural log ln is the logarithm to the base e. Whereas the log is the logarithm to the base 10.