Q. 15 - Calculus

Question

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

Step-by-Step Solution

Answer

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

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,

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