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 ?
