Solve ∫ X Ln(x) Dx: Step-by-Step Integral Solution

markdown # Solve ∫ x ln(x) dx: Step-by-Step Integral Solution Hello there! Today, we're diving into the world of calculus to solve the integral of *x ln(x)* with respect to *x*. This is a classic problem that often appears in calculus courses and involves a technique called integration by parts. Don't worry, we'll break it down step-by-step to make it crystal clear. We'll start with the direct answer for those who need it quickly, and then we'll delve into a detailed explanation of the process. ## Correct Answer The integral of *x ln(x)* with respect to *x* is: **(x^2 / 2) * ln(x) - (x^2 / 4) + C** where *C* is the constant of integration. ## Detailed Explanation Now, let's understand how we arrive at this answer. Integrating *x ln(x)* requires a technique known as ***integration by parts***. This method is particularly useful when you have a product of two functions, as we do here with *x* and *ln(x)*. ### Key Concepts Before we jump into the solution, let's refresh the key concepts: 1. ***Integration by Parts:*** This technique comes from the product rule for differentiation. The formula for integration by parts is: ∫ u dv = uv - ∫ v du Where: * *u* is a function we choose to differentiate. * *dv* is a function we choose to integrate. * *du* is the derivative of *u*. * *v* is the integral of *dv*. 2. ***Choosing u and dv:*** The key to successful integration by parts is choosing appropriate *u* and *dv*. A helpful guideline is the acronym **LIATE**: * **L** - Logarithmic functions (ln(x), log_b(x)) * **I** - Inverse trigonometric functions (arctan(x), arcsin(x)) * **A** - Algebraic functions (x, x^2, polynomials) * **T** - Trigonometric functions (sin(x), cos(x)) * **E** - Exponential functions (e^x, a^x) The function that appears *earlier* in this list is generally a good choice for *u*, as it simplifies upon differentiation. Let's apply these concepts to our integral: ∫ *x ln(x)* dx ### Step-by-Step Solution: 1. ***Identify u and dv:*** Looking at our integral ∫ *x ln(x)* dx, we have two functions: *x* (algebraic) and *ln(x)* (logarithmic). According to LIATE, logarithmic functions come before algebraic functions. Therefore, we choose: * u = ln(x) * dv = x dx 2. ***Find du and v:*** Now we need to find *du* (the derivative of *u*) and *v* (the integral of *dv*): * du = d/dx [ln(x)] = (1/x) dx * v = ∫ x dx = (x^2 / 2) 3. ***Apply the Integration by Parts Formula:*** Recall the formula: ∫ u dv = uv - ∫ v du Substitute the values we found: ∫ *x ln(x)* dx = ln(x) * (x^2 / 2) - ∫ (x^2 / 2) * (1/x) dx 4. ***Simplify the New Integral:*** Let's simplify the integral on the right side: ∫ (x^2 / 2) * (1/x) dx = ∫ (x / 2) dx 5. ***Solve the Simplified Integral:*** Now we can easily integrate (x / 2): ∫ (x / 2) dx = (1/2) ∫ x dx = (1/2) * (x^2 / 2) = (x^2 / 4) 6. ***Substitute Back into the Integration by Parts Formula:*** Now, plug the result back into our equation: ∫ *x ln(x)* dx = ln(x) * (x^2 / 2) - (x^2 / 4) 7. ***Add the Constant of Integration:*** Finally, we add the constant of integration, *C*, because the indefinite integral represents a family of functions: ∫ *x ln(x)* dx = (x^2 / 2) * ln(x) - (x^2 / 4) + C So, there you have it! The integral of *x ln(x)* with respect to *x* is (x^2 / 2) * ln(x) - (x^2 / 4) + C. Let's break down why each step is crucial. ### Deeper Dive into the Steps: * ***Choosing u and dv (LIATE):*** The choice of *u* and *dv* is critical. If we had chosen *u = x* and *dv = ln(x) dx*, we would have needed to find the integral of *ln(x)*, which is a bit more complex. By choosing *u = ln(x)*, we simplified the differentiation step, making the subsequent integration easier. * ***Integration by Parts Formula:*** The formula ∫ u dv = uv - ∫ v du is the cornerstone of this technique. It allows us to transform a complex integral into a simpler one, provided we make the right choice for *u* and *dv*. * ***Simplifying the Integral:*** After applying the formula, we often end up with a new integral, ∫ v du. The goal is that this new integral is easier to solve than the original. In our case, ∫ (x / 2) dx was much simpler than ∫ *x ln(x)* dx. * ***Constant of Integration:*** Never forget the constant of integration, *C*, for indefinite integrals! This is because the derivative of a constant is zero, so there are infinitely many antiderivatives that differ by a constant. ### Common Mistakes to Avoid: 1. ***Incorrect Choice of u and dv:*** A wrong choice can lead to a more complicated integral. Always consider LIATE or similar rules. 2. ***Forgetting the Constant of Integration:*** This is a common mistake. Always add *C* to the final answer of an indefinite integral. 3. ***Incorrect Differentiation or Integration:*** Double-check your derivatives and integrals, especially in the intermediate steps. A small error can propagate through the entire solution. 4. ***Algebraic Errors:*** Be careful with your algebra, especially when simplifying the new integral and substituting back. Let's consider another example to solidify your understanding: ∫ x * e^x dx 1. ***Identify u and dv:*** * u = x (Algebraic) * dv = e^x dx (Exponential) 2. ***Find du and v:*** * du = dx * v = ∫ e^x dx = e^x 3. ***Apply the Formula:*** ∫ x * e^x dx = x * e^x - ∫ e^x dx 4. ***Solve the Simplified Integral:*** ∫ e^x dx = e^x 5. ***Substitute and Add C:*** ∫ x * e^x dx = x * e^x - e^x + C Notice how choosing *u = x* simplified the problem, whereas choosing *u = e^x* would have made it more complicated. The key is practice! The more you work through integration by parts problems, the better you'll become at choosing the appropriate *u* and *dv*. ## Conclusion Key Takeaways: * ***Integration by parts*** is essential for integrals involving the product of functions. * The formula is: ∫ u dv = uv - ∫ v du * Use **LIATE** (Logarithmic, Inverse trigonometric, Algebraic, Trigonometric, Exponential) to help choose *u* and *dv*. * Always add the constant of integration, **C**, for indefinite integrals. * Practice is key to mastering this technique. I hope this detailed explanation has helped you understand how to solve the integral of *x ln(x)*. If you have any more questions or want to explore other calculus problems, feel free to ask! Happy integrating!