Introduction to the integral of ln x 2 

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. 

Integral calculators 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 squared. You will also understand how to compute ln x 2 integral by using different integration techniques.

What is the integral of lnx 2?

The integral of the ln x square is an antiderivative of the (ln x)^2 function which is equal to x(ln x)^2 - 2xln x+2x+c. It is also known as the reverse derivative of the ln x 2 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 square formula

The formula of antiderivative of ln x 2 contains integral sign, coefficient of integration and the function as sine. It is denoted by ∫(ln x)dx. In mathematical form, the integral of lnx^2 is:

$∫(\ln x)^2dx = x(\ln x)^2 - 2x\ln x+2x+c$

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

How to calculate the integral ln x 2?

The integral of ln x 2 is its antiderivative, which can be calculated using different integration techniques. In this article, we will discuss how to calculate integral of ln x squared by using:

1. U-substitution method
2. Integration by parts

Integral of ln x squared 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 lnx 2 by using u-substitution.

Proof of Integral of lnx 2 by using substitution method

To proof the integral of (lnx)^2 by using substitution method, suppose that:

$I=\int(\ln x)^2dx$

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.

$I=\int u^2e^udu$

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, 

$f(u)=u^2$ and $g(u)=e^u$

Then, by using above formula, 

$I=u^2\int e^udu-\int [\frac{d}{du}(u^2)\int e^udu]du$

$I=u^2e^u-2\int ue^udu$

$I=u^2e^u-2(ue^u-\int e^udu)$

$I=u^2e^u-2ue^u+2e^u+c

Now substituting the value of u, we get

$I=x(\ln x)^2-2x\ln x+2x+c$

Hence we have verified the antiderivative of ln x 2 by using the u-substitution method. You can also use the u-substitution calculator to make these calculations easier.

Integral of lnx^2 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 lnx 2 by using integration by parts.

Proof of integral of ln x 2 by using integration by parts

To integrate the function ln(x)^2 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, 

$f(x) = (\ln x)^2$

$g(x) =1$ 

Using these values in the integration by parts formula calculator, 

$I=(\ln x)^2\int 1dx-\int[\frac{d}{dx}(\ln x)^2\int 1dx]dx$

Integrating, 

$I=x(\ln x)^2-2\int \ln x\left(\frac{1}{x}\times x\right)dx$

$I=x(\ln x)^2-2\int \ln xdx$

Since the integral of ln x is equal to xln x - x, then 

$I=x(\ln x)^2-2[x\ln x-x]+c$

$I=x(\ln x)^2-2x\ln x+2x+c$

Hence the antiderivative of ln x 2 is x (ln x)^2 - 2xln x+2x+c, where c is a constant.

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.