Question

Verify that$\int \ln xdx=x(\ln x-1)+C$(Do not try to solve the integral from scratch.

Short answer

It is verified that$\int \ln xdx=x(\ln x-1)+C.$

Step-by-step solution

Step 1. Given information.

The given integral is$\int \ln xdx=x(\ln x-1)+C.$

Step 2. Verification.

Now, we can write,

$$\begin{gathered} \frac{d}{dx}x(\ln x-1) \\ =1(\ln x-1)+x\left(\frac{1}{x}-0\right) \\ =\ln x-1+1 \\ =\ln x \end{gathered}$$

So,

$x(\ln x-1)\text{is an anti-derivative of}\ln x\text{.}$

Therefore,

$\int \ln xdx=x(\ln x-1)+C$