Integrate the function w..r. to x : log x
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To integrate log x with respect to x,use Integration by parts
![\int U.V dx= U\int V dx-\int[\frac{dU}{dx}\int V dx]dx\\\\](https://tex.z-dn.net/?f=+%5Cint+U.V+dx%3D+U%5Cint+V+dx-%5Cint%5B%5Cfrac%7BdU%7D%7Bdx%7D%5Cint+V+dx%5Ddx%5C%5C%5C%5C "\int U.V dx= U\int V dx-\int[\frac{dU}{dx}\int V dx]dx\\\\")
![\int1. log\:x dx= log x\int 1 dx-\int[\frac{d log x}{dx}\int 1 dx]dx\\\\=x log\:x-\int[\frac{1}{x} .x] dx\\\\\\=x log x-\int 1.dx\\\\\\\int log\: x \:dx= x log x- x+C](https://tex.z-dn.net/?f=%5Cint1.+log%5C%3Ax+dx%3D+log+x%5Cint+1+dx-%5Cint%5B%5Cfrac%7Bd+log+x%7D%7Bdx%7D%5Cint+1+dx%5Ddx%5C%5C%5C%5C%3Dx+log%5C%3Ax-%5Cint%5B%5Cfrac%7B1%7D%7Bx%7D+.x%5D+dx%5C%5C%5C%5C%5C%5C%3Dx+log+x-%5Cint+1.dx%5C%5C%5C%5C%5C%5C%5Cint+log%5C%3A+x+%5C%3Adx%3D+x+log+x-+x%2BC "\int1. log\:x dx= log x\int 1 dx-\int[\frac{d log x}{dx}\int 1 dx]dx\\\\=x log\:x-\int[\frac{1}{x} .x] dx\\\\\\=x log x-\int 1.dx\\\\\\\int log\: x \:dx= x log x- x+C")
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