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?aSolution: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=uv–∫vdu——

i

Let x=log(x), dv=duDifferentating on both sides, we getdx=1

x, v=uSubstuting the values of u and dx in equation

i

, we get∫log(x) dx=x log(x)–∫x.d

dxlog(x)dx∫log(x) dx=x log(x)–∫x.1

xdx∫log(x) dx=x log(x)–∫x.1

xdx∫log(x) dx=x log(x)–∫dx∫log(x) dx=x log(x)–∫d x∫log(x) dx=x log(x)–uTaking 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.

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