 # Integral of Natural Log | How to integrate ln u?

## How to integrate ln u?

According to the given question, we are asked to find the integral of ln u.

Solution: We know that log x is a logarithmic function.

It is not possible to integrate any logarithmic function directly.

So, we will use integration by parts to integrate ∫ log(x) dx.

According to integration by parts,

∫u dv=uv-∫vdu——i

Let x = log(x), dv = du

Differentiating on both sides, we get

How to integrate ln x | What is the integral of ln x?Solution: We know that log x is a logarithmic function and it is not possible to intergrate any logarithmic function directly. So, we will use intergration by parts to integrate log(x) dx. According to integration by parts, u dv=uvvdu——
i
Let x= log(x), dv=du Differentating on both sides, we get dx=1
x
, v=u
Substuting the values of u and dx in equation
i
, we get
log(x) dx=x log(x)x.d
dx
log(x)dx
log(x) dx=x log(x)x.1
x
dx
log(x) dx=x log(x)x.1
x
dx
log(x) dx=x log(x)dx log(x) dx=x log(x)d x log(x) dx=x log(x)u Taking x common, we get log(x) dx=x(log(x))1)+C The integral of logx=x(log(x))1)+C, where C is the integration constant.

## Summary

To solve ln u, we have used integration by parts and the integral of log u = u(log(u))-1)+C, where C is the integration constant.