# Integration of x ln(x) Explained Simply
Hello there! Today, we're going to tackle a common integration problem: ∫x ln(x) dx. Don't worry, we'll break it down step-by-step so you can understand exactly how to solve it. We'll provide the direct answer first, followed by a detailed explanation to ensure you grasp the underlying concepts.
## Correct Answer
The integral of x ln(x) is **(x^2 / 2) ln(x) - x^2 / 4 + C**, where C is the constant of integration.
## Detailed Explanation
To integrate x ln(x), we'll use a technique called *integration by parts*. This method is particularly useful when you have a product of two functions, as we do here (x and ln(x)). Integration by parts is essentially the reverse of the product rule for differentiation. Let's dive in!
### Integration by Parts Formula
The integration by parts formula is:
∫u dv = uv - ∫v du
Where:
* u and v are functions of x.
* du is the derivative of u.
* dv is the derivative of v.
### Choosing u and dv
The key to successful integration by parts is choosing the right functions for u and dv. A helpful acronym to guide this choice is **LIATE**:
* **L**ogarithmic functions (like ln(x))
* **I**nverse trigonometric functions (like arctan(x))
* **A**lgebraic functions (like x, x^2)
* **T**rigonometric functions (like sin(x), cos(x))
* **E**xponential functions (like e^x)
The function that appears *higher* on this list is generally a good choice for *u*, and the rest goes to *dv*. In our case, we have ln(x) (logarithmic) and x (algebraic). Logarithmic comes before algebraic in LIATE, so:
* u = ln(x)
* dv = x dx
### Finding du and v
Now, we need to find du and v:
* du = d/dx [ln(x)] dx = (1/x) dx
* v = ∫dv = ∫x dx = x^2 / 2
### Applying the Integration by Parts Formula
Plug u, dv, du, and v into the integration by parts formula:
∫x ln(x) dx = ln(x) * (x^2 / 2) - ∫(x^2 / 2) * (1/x) dx
Simplify the integral:
∫x ln(x) dx = (x^2 / 2) ln(x) - (1/2) ∫x dx
### Evaluating the Remaining Integral
Now we need to integrate x, which is a straightforward power rule integration:
∫x dx = x^2 / 2 + C'
(We'll combine the constant of integration later.)
### Putting It All Together
Substitute this result back into our equation:
∫x ln(x) dx = (x^2 / 2) ln(x) - (1/2) * (x^2 / 2) + C
Simplify:
∫x ln(x) dx = (x^2 / 2) ln(x) - x^2 / 4 + C
And that's our final answer!
### Example Problem
Let's do another similar example to solidify your understanding.
Evaluate the definite integral: ∫[1 to e] x ln(x) dx
#### Steps:
1. We already know the indefinite integral: (x^2 / 2) ln(x) - x^2 / 4 + C
2. Now, we evaluate this expression at the upper limit (e) and the lower limit (1).
3. Subtract the value at the lower limit from the value at the upper limit.
#### Evaluation:
* At x = e: ((e^2) / 2) ln(e) - (e^2) / 4 = (e^2 / 2) * 1 - e^2 / 4 = e^2 / 2 - e^2 / 4 = e^2 / 4
* At x = 1: ((1^2) / 2) ln(1) - (1^2) / 4 = (1 / 2) * 0 - 1 / 4 = -1 / 4
#### Definite Integral Result:
∫[1 to e] x ln(x) dx = (e^2 / 4) - (-1 / 4) = (e^2 / 4) + (1 / 4) = (e^2 + 1) / 4
### Key Concepts
* ***Integration by Parts:*** A technique used to integrate the product of two functions. The formula is ∫u dv = uv - ∫v du.
* ***LIATE Rule:*** A mnemonic to help choose u and dv in integration by parts (Logarithmic, Inverse trigonometric, Algebraic, Trigonometric, Exponential).
* ***Power Rule for Integration:*** ∫x^n dx = (x^(n+1)) / (n+1) + C, where n ≠ -1.
* ***Logarithmic Function ln(x):*** The natural logarithm of x, its derivative is 1/x, and ln(1) = 0, ln(e) = 1.
* ***Definite Integral:*** An integral with defined upper and lower limits, giving a numerical value.
* ***Indefinite Integral:*** An integral without defined limits, resulting in a function plus a constant of integration (C).
### Common Mistakes to Avoid
* **Incorrectly applying the Integration by Parts formula:** Double-check the formula and ensure correct substitution of u, dv, du, and v.
* **Choosing the wrong u and dv:** Using LIATE can be a good starting point, but sometimes, another choice may simplify the integration process.
* **Forgetting the Constant of Integration (C):** Always include + C when evaluating indefinite integrals.
* **Algebra Errors:** Be cautious with algebraic manipulations, especially when dealing with fractions and exponents.
* **Incorrectly evaluating Definite Integrals:** Make sure to substitute the limits of integration correctly and subtract the values in the right order.
## Key Takeaways
* To integrate x ln(x), use integration by parts.
* Choose u = ln(x) and dv = x dx based on the LIATE rule.
* The integral of x ln(x) is (x^2 / 2) ln(x) - x^2 / 4 + C.
* Remember the integration by parts formula: ∫u dv = uv - ∫v du.
* For definite integrals, evaluate the result at the upper and lower limits and subtract.
I hope this explanation has made the integration of x ln(x) clear and understandable! If you have any more questions, feel free to ask.
