When encountering integrals of the form ∫x ln x dx, many students and professionals alike may feel a sense of unease or uncertainty. However, with a clear understanding of the underlying principles and a step-by-step approach, solving these integrals can become a straightforward and manageable task. In this article, we will delve into the world of x ln x integrals, exploring the theoretical foundations and providing a practical, 5-step guide to solving these integrals with ease.
Key Points
- Recognize the integral of x ln x as a classic example of an integral that requires integration by parts.
- Choose u and dv wisely to simplify the integration process.
- Apply the integration by parts formula: ∫u dv = uv - ∫v du.
- Utilize the fact that the integral of 1/x is ln|x| to solve for the remaining integral.
- Combine the results to obtain the final antiderivative.
Understanding the Theoretical Foundations
To tackle the integral ∫x ln x dx, it’s essential to have a solid grasp of integration by parts, a fundamental technique in calculus. The formula for integration by parts is ∫u dv = uv - ∫v du, where u and dv are functions of x. By strategically selecting u and dv, we can simplify the integration process and make it more manageable. In the case of ∫x ln x dx, we can let u = ln x and dv = x dx, which implies du = 1/x dx and v = (1⁄2)x^2.
Step 1: Select u and dv
The first step in solving the integral ∫x ln x dx is to choose suitable functions for u and dv. As mentioned earlier, we let u = ln x and dv = x dx. This selection is not arbitrary; rather, it’s based on the fact that the derivative of ln x is 1/x, and the integral of x dx is (1⁄2)x^2. By making this choice, we can simplify the subsequent calculations and make the integration process more straightforward.
Step 2: Apply the Integration by Parts Formula
With u and dv selected, we can now apply the integration by parts formula: ∫u dv = uv - ∫v du. Substituting u = ln x and dv = x dx, we get ∫x ln x dx = (1⁄2)x^2 ln x - ∫(1⁄2)x^2 (1/x) dx. Simplifying the integral on the right-hand side, we have ∫x ln x dx = (1⁄2)x^2 ln x - (1⁄2)∫x dx.
| Function | Derivative | Integral |
|---|---|---|
| ln x | 1/x | x ln x - x |
| x | 1 | (1/2)x^2 |
Step 3: Evaluate the Remaining Integral
The integral ∫x dx is a straightforward one, and its evaluation is (1⁄2)x^2 + C, where C is the constant of integration. Substituting this result back into the equation from Step 2, we get ∫x ln x dx = (1⁄2)x^2 ln x - (1⁄2)((1⁄2)x^2) + C. Simplifying further, we have ∫x ln x dx = (1⁄2)x^2 ln x - (1⁄4)x^2 + C.
Step 4: Combine the Results
Combining the results from the previous steps, we can now write the complete antiderivative of ∫x ln x dx as (1⁄2)x^2 ln x - (1⁄4)x^2 + C. This expression represents the indefinite integral, and it’s essential to include the constant of integration, C, to ensure the result is accurate and complete.
Step 5: Verify the Result (Optional)
Although not strictly necessary, verifying the result by differentiating the antiderivative can provide an added layer of confidence in the solution. Differentiating (1⁄2)x^2 ln x - (1⁄4)x^2 + C with respect to x, we get x ln x, which is indeed the original function. This verification step can help catch any potential errors and ensure the correctness of the result.
What is the primary technique used to solve ∫x ln x dx?
+The primary technique used to solve ∫x ln x dx is integration by parts, which involves selecting suitable functions for u and dv and applying the formula ∫u dv = uv - ∫v du.
How do I choose u and dv for the integral ∫x ln x dx?
+For the integral ∫x ln x dx, we can let u = ln x and dv = x dx. This selection is based on the fact that the derivative of ln x is 1/x, and the integral of x dx is (1⁄2)x^2.
What is the final antiderivative of ∫x ln x dx?
+The final antiderivative of ∫x ln x dx is (1⁄2)x^2 ln x - (1⁄4)x^2 + C, where C is the constant of integration.