Integral of (x \ln x), a common yet significant integral, arises in various mathematical disciplines and applications. Its close association with the logarithmic function and its derivative, (ln(x)), makes it a fundamental concept in calculus. Furthermore, the integral finds application in probability theory, where it serves as the distribution function of the exponential distribution. Its connection to the exponential function, (e^x), highlights its importance in modeling growth and decay phenomena.
Cracking the Integral: Tackling x ln x dx
When you encounter an integral of the form x ln x dx, don’t panic! Follow this structured approach to conquer it:
Integration by Parts
In situations like this, integration by parts is your ally. Split the integrand into two functions, u and dv:
- Let u = ln x and dv = x dx.
Calculate du and v:
- du = 1/x dx
- v = x^2/2
Using the formula
∫ u dv = uv – ∫ v du,
implement integration by parts:
∫ x ln x dx = (x^2/2) ln x – ∫ (x^2/2) * (1/x) dx
Simplify the integral:
∫ x ln x dx = (x^2/2) ln x – (1/2) ∫ x dx
Basic Integration
Now, you’re left with a basic integral:
∫ x dx = x^2/2
Putting It All Together
Substitute the result back into the original equation:
∫ x ln x dx = (x^2/2) ln x – (1/2) * (x^2/2)
Simplify:
∫ x ln x dx = (x^2/2) ln x – x^2/4
That’s it! You’ve successfully integrated x ln x dx.
Question 1:
How can integration by parts be used to evaluate the integral of x ln x dx?
Answer:
Integration by parts, also known as integration by reduction formula, involves splitting an integral into two parts: one part that can be integrated easily, and another part that can be reduced to a simpler integral. For the integral of x ln x dx, u is set to ln x and dv is set to x dx. Applying the integration by parts formula, we get:
∫ x ln x dx = (ln x) ∫ x dx - ∫ [d/dx (ln x)] ∫ x dx
Simplifying further:
∫ x ln x dx = ln x * (x^2 / 2) - ∫ (1 / x) * (x^2 / 2) dx
∫ x ln x dx = (x^2 * ln x) / 2 - (x^2 / 4) + C
Where C is the constant of integration.
Question 2:
What is the significance of the constant of integration in the evaluation of the integral of x ln x dx?
Answer:
The constant of integration represents the indefinite nature of the integral. Any derivative of a constant is zero, so adding a constant to the solution of an integral does not change its derivative. This means that the solution to an indefinite integral is always a family of functions, each differing by a constant. The specific value of the constant of integration depends on the specific boundary conditions or initial conditions of the problem being solved.
Question 3:
How does the integrand of the integral of x ln x dx behave as x approaches zero and infinity?
Answer:
As x approaches zero, the integrand x ln x approaches negative infinity. This is because ln x approaches negative infinity as x approaches zero from the positive side. As x approaches infinity, the integrand x ln x approaches infinity. This is because both x and ln x approach infinity as x approaches infinity.
Well, there you have it, folks! The intricacies of the “x ln x dx” integral have been unveiled. Remember, it’s not just about the answer, but about the journey of understanding the techniques involved. Don’t be afraid to tackle other integrals that come your way. The more you practice, the more confident you’ll become. Thanks for joining me on this mathematical adventure. Keep exploring, keep learning, and I’ll catch you next time for another dose of integral enlightenment!