Evaluate: x d x - Vidyadip

Question

Evaluate: $\int \log x d x$

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Answer

We can write log (x) = 1. log (x) so
$\int \log x d x=\int 1 \cdot \log (x) d x$
We can solve it by using integration by parts , For this we take log(x)as first function and 1 as second function.
$\int 1 \cdot \log (x) d x$
$\log ( x ) \int 1 . d x-\int\left(\frac{d}{d x}\right)(\log x) \int 1 . d x$
$=\log (x) \cdot x-\int \frac{1}{x} \times x d x$
$=\log (x) \cdot x-\int 1 \cdot d x$
= log (x) x - x + c
= x (log(x) - 1) + c
Hence, the value of $\int \log (x) d x$ is $(\log (x)-1)+c$

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