# X*ln(x) = 1/2x^2*ln(x) - 1/4*x^2 not sure how they got it

• mr_coffee
In summary, the equation X*ln(x) = 1/2x^2*ln(x) - 1/4*x^2 is derived using properties of logarithms and basic algebraic manipulation. To solve the equation, techniques such as factoring, substitution, or graphing can be used, as well as numerical methods like Newton's method. The possible solutions for this equation depend on the value of x and can be found by setting each side of the equation equal to 0 and solving for x. In mathematics, this equation is significant as it demonstrates the relationship between two functions and the manipulation of logarithmic and exponential functions. It can also be applied to real-world problems in various fields.

#### mr_coffee

Hello everyone!

I'm doing a line integral but I'm confused on the integration method used:

I got to integral 1 to 3 2*(x^4+xln(x) ); But they then have:
x^5/5 + 1/2x^2 * ln(x) - 1/4x^2 and I'm not sure how they got the last part, 1/2x^2 * ln(x) - 1/4x^2

Thanks!

u = x isn't what they did I don't think, and its been awhile so I'm rusty on integration XD

The 'hard' part is the integral of x*ln(x). And they just integrated by parts.

ahh thank you

## 1. What is the equation X*ln(x) = 1/2x^2*ln(x) - 1/4*x^2 and how was it derived?

The equation X*ln(x) = 1/2x^2*ln(x) - 1/4*x^2 is a mathematical expression that equates two functions, X*ln(x) and 1/2x^2*ln(x) - 1/4*x^2. It was derived using properties of logarithms and basic algebraic manipulation.

## 2. How do you solve the equation X*ln(x) = 1/2x^2*ln(x) - 1/4*x^2?

To solve this equation, you can use techniques such as factoring, substitution, or graphing to find the values of x that satisfy the equation. You can also use numerical methods such as Newton's method to approximate the solutions.

## 3. What are the possible solutions for the equation X*ln(x) = 1/2x^2*ln(x) - 1/4*x^2?

The possible solutions for this equation depend on the value of x. Some values of x may not have real solutions, while others may have multiple solutions. The solutions can be found by setting each side of the equation equal to 0 and solving for x.

## 4. What is the significance of the equation X*ln(x) = 1/2x^2*ln(x) - 1/4*x^2 in mathematics?

This equation is significant in mathematics because it shows the relationship between two different functions, X*ln(x) and 1/2x^2*ln(x) - 1/4*x^2. It also demonstrates how logarithmic and exponential functions can be manipulated and equated to each other.

## 5. Can this equation be applied to real-world problems?

Yes, this equation can be applied to real-world problems in various fields such as physics, chemistry, and economics. It can be used to model growth and decay, as well as in optimization and probability calculations.