Integral of Ln(x+1)

Integral of ln(x+1) along with its formula and proof with examples. Also learn how to calculate integration of ln(x+1) with step by step examples.

Alan Walker-

Published on 2023-04-27

Introduction to the integral of ln(x+1)

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 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+1). You will also understand how to compute the ln (x+1) integral by using different integration techniques.

What is the integral of ln (x+1)?

The integral of ln(x+1) is an antiderivative of the ln(x+1) function which is equal to (x+1)ln (x+1) - x +c. It is also known as the reverse derivative of a natural log of x+1.

The function ln (x+1) is a logarithm to the base e, where e is the Euler’s number equal to 2.7182. The function ln (x+1) is also known as the natural log of x+1. 

Integral of ln(x+1) formula

The formula of the integral of ln contains the integral sign, coefficient of integration and the function as sine. It is denoted by ∫(ln (x+1))dx. In mathematical form, the integral of sin x is:

 $$∫\ln(x+1)dx = (x+1)\ln(x+1) - x + c$$

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

How to calculate the integral of ln(x+1)?

The integral of ln (x+1) is its antiderivative that can be calculated by using different integration techniques. You can also use an indefinite integration solver to find this integral. In this article, we will discuss how to calculate the integral of ln x+1 by using:

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

Integral of log (x+1) 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 that is complex to be calculated by usual integration. Let’s discuss calculating the integral of ln (x+1) by using the u-substitution.

Proof of Integral of natural log by using substitution method

To prove the integral of log (x+1) by using the substitution method, suppose that:

$$I=\int \ln(x+1)dx$$

Suppose that,

$$u=\ln (x+1)$$


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


$$(x+1)du = dx$$



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, 

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

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


Now substituting the value of u, we get


Thus, the integral of log (x+1) is equal to (x+1)ln(x+1)-x+c. You can also use the u-substitution integration calculator to verify this integral.

Integral of ln(x+1) 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. Let’s discuss calculating the integral of ln(x+1) by using integration by parts.

Proof of integral of ln (x+1) by using integration by parts

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

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

Suppose that, 

$$f(x) = \ln(x+1)$$

$$g(x) =1$$ 

Using these values in the above formula of integral by parts calculator

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


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

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

More simplification, 


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

Integrating the remaining terms,




Hence the integral of ln (x+1) is (x+1)ln (x+1) - x+c, where c is a constant.

Integral of natural log of x+1 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. Our area under the curve calculator may help you in calculating area under a curve. The definite integral can be written as:

$$∫^b_a f(x) dx = F(b) – F(a)$$

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

$$∫^1_0 \ln(x+1) dx = \left|(x+1)\ln(x+1) - x\right|^1_0$$ 

Now, substitute the limit in the given function.

$$∫^1_0 \ln(x+1) dx = [(1+1)\ln (2) - 1]-[(0+1)\ln 0 -0]$$

$$∫^1_0 \ln(x+1) dx = 2\ln(2)-1$$

Since ln 2 is equal to 0.7, therefore, 

$$∫^1_0 \ln(x+1) dx = 1.3862-1=0.3862$$

Which is the calculation of the definite integral of log x+1.

Frequently Asked Questions

How do you evaluate the integral of ln(x+1)?

The antiderivative or the integral of the natural log of (x+1) is equal to the difference of (x+1)ln (x+1) and x. It can be obtained by using different integration techniques such as the u-substitution, integration by parts, and the definite integral of ln (x+1).

What is the domain of ln(x+1)?

The domain of the logarithmic function ln(x+1) is all positive real numbers. It is because the function ln(x+1) is defined for only positive values of x. 

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