Question

Evaluate $\int (1+\ln x)dx$
(where $C$ is constant of integration)

• A. $\ln x+C$
• B. $x\ln x+C$
• C. $x+C$
• D. $x+\ln x+C$

Answer 1

The correct option is B $x\ln x+C$

Let $I=\int (1+\ln x)dx$
$\Rightarrow I=\int (x\cdot \frac{1}{x}+\ln x)dx$

Clearly it is of the form $\int \left(x{f}^{\prime }\right(x)+f(x\left)\right)dx=xf\left(x\right)$
where $f\left(x\right)=\ln x$.

Hence, $I=x\ln x+C$.