The integral of 1/x log(x) or lnx is the antiderivative of the function 1/x log(x) or lnx, which is the function that, when differentiated, gives the original function as its result. In other words, it is the inverse operation of differentiation.
Yes, the integral 1/x log(x) or lnx can be solved analytically using integration by parts. This method involves breaking down the integral into simpler parts and using a formula to solve each part. However, the resulting integral may still be difficult to evaluate without the use of numerical methods.
The integral 1/x log(x) or lnx is used in various fields of science and engineering, such as physics, chemistry, and economics. It is used to calculate quantities such as work, energy, and population growth, which are essential in understanding natural phenomena and making predictions.
The domain of the function 1/x log(x) or lnx is all positive real numbers greater than zero, since the natural logarithm of a negative number is undefined. The range of the function is all real numbers, as the natural logarithm can take on any real value.
One important property to keep in mind is that the integral 1/x log(x) or lnx is not defined at x = 0, as the function 1/x becomes undefined at this point. Another property is that the integral is considered an improper integral, meaning that it does not converge to a finite value and may require special techniques to evaluate.