# What is the Integration of Log x dx ?

Here you will learn what is the integration of log x dx with respect to x and examples based on it.

Let’s begin –

## Integration of Log x

The integration of log x with respect to x is x(log x) – x + C.

where C is the integration Constant.

i.e. $$\int$$ log x dx = x(log x) – x + C

Proof :

We will use integration by parts formula to prove this,

Let I = $$\int$$ log x .1 dx

where log x is the first function and 1 is the second function according to ilate rule.

I = log x . {$$\int$$ 1 dx} – $$\int$$ { $$d\over dx$$ (log x) . $$\int$$ 1 dx } dx

I = (log x) x – $$\int$$ $$1\over x$$.x dx

= x (log x) – $$\int$$ 1 dx

= x (log x) – x + C

Hence, $$\int$$ log x = x (log x) – x + C

Example : Evaluate : $$(log x)^2$$ dx

Solution : We have,

I = $$(log x)^2$$ . 1 dx, Then ,

where $$(log x)^2$$ is the first function and 1 is the second function according to ilate rule,

I = $$(log x)^2$$ { $$\int$$ 1 dx} – $$\int$$ {$$d\over dx$$ $$(log x)^2$$ . $$\int$$ 1 dx } dx

= $$(log x)^2$$ x – $$\int$$ 2 log x . $$1\over x$$ . x dx

= x $$(log x)^2$$ – 2 $$\int$$ log x .1 dx

$$\implies$$ I = x $$(log x)^2$$ – 2[ log x { $$\int$$ 1 dx } – $$\int$$ { $$d\over dx$$ (log x) $$\int$$ 1 dx } dx ]

$$\implies$$ I = x $$(log x)^2$$ – 2 { (log x) x – $$\int$$ $$1\over x$$ x dx }

Hence, I = x( $$(log x)^2$$ – 2 (x log x – x) + C

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