Question

Решить: int[(lnx)/x dx]

Answer 1

Int[lnx]/x dx = int[lnx] d(lnx), т.к. d(lnx) = 1/x dx
int[lnx] d(lnx) = (lnx)^2/2

Answer 2

To solve the integral ∫(ln(x)/x) dx, we can use integration by parts. The formula for integration by parts is ∫(u dv) = uv - ∫(v du), where u and v are differentiable functions of x.

Let's choose u = ln(x) and dv = dx.

Then, we have du = (1/x) dx and v = ∫dx = x.

Now, we can apply the integration by parts formula:

∫(ln(x)/x) dx = ∫u dv = uv - ∫v du = ln(x) * x - ∫x * (1/x) dx = x * ln(x) - ∫1 dx = x * ln(x) - x + C

where C is the constant of integration.

So, the solution to the integral is:

∫(ln(x)/x) dx = x * ln(x) - x + C