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


Related Questions

What is the integration of log cos x dx ?

What is the integration of log 1/x ?

What is the integration of 1/x log x ?

What is the integration of x log x dx ?

Leave a Comment

Your email address will not be published.