What's the integral of ln(x)?

The integral of ln(x) is xln(x) - x + C, where C is the constant of integration.

To find the integral of ln(x), we use integration by parts. Let u = ln(x) and dv = dx. Then du/dx = 1/x and v = x. Using the formula for integration by parts, we have:

∫ln(x) dx = xln(x) - ∫x(1/x) dx
= xln(x) - ∫dx
= xln(x) - x + C

where C is the constant of integration.

We can check our answer by differentiating it using the product rule. Let y = xln(x) - x + C. Then

dy/dx = d/dx(xln(x)) - d/dx(x) + d/dx(C)
= (x d/dx(ln(x)) + ln(x)) - 1 + 0
= ln(x) + 1/x - 1
= ln(x)/x

which is the integrand we started with, so our answer is correct.

Note that the integral of ln(x) is only defined for x > 0, since ln(x) is not defined for x ≤ 0.