why is integral lnx integration by parts

[opal path]

Just curious why we cant integrate it as 1/x, im in calc 2 so this is definitely unnecessary reasoning but im just curious

[iron agate]

Because the derivative of 1/x is equal to -1/x^2

And -1/x^2 is not the same function as ln

Since -1/x^2 is equal to -1 at x=1
But ln(1)=0

The integral of ln(x) (aka the antiderivative) must be a function such that its derivative is exactly ln(x)

You are confusing with the fact that the derivative of ln(x) is 1/x

[mild hatch]

The integral ln(x) , dx is by integration by parts because it is the integral of a product of two functions, one of which is ln(x) and the other of which is 1. We can let u = ln(x) and dv = dx, so du = dx/x and v = x. Then, using the integration by parts formula,

[mild hatch]

Now, you can use the integration by parts formula:

∫ln(x)dx = uv - ∫v du

∫ln(x)dx = ln(x) * x - ∫x * (1/x)dx

Now, simplify the expression:

∫ln(x)dx = x * ln(x) - ∫dx

The integral of dx is simply x:

∫ln(x)dx = x * ln(x) - x + C

Where C is the constant of integration.

So, the integral of ln(x) with respect to x is x * ln(x) - x + C.