Question

Evaluate $\int x\ln xdx$
(where $C$ is constant of integration)

• A. $\frac{x^2\ln x}{2}-\frac{x^2}{4}+C$
• B. $\frac{x^2\ln x}{2}+\frac{x^2}{4}+C$
• C. $-\frac{x^2\ln x}{2}-\frac{x^2}{4}+C$
• D. $-\frac{x^2\ln x}{2}+\frac{x^2}{4}+C$

Answer 1

The correct option is A $\frac{x^2\ln x}{2}-\frac{x^2}{4}+C$

Let $I=\int x\ln xdx$
Using ILATE rule, we get $\Rightarrow I=\ln x\cdot \frac{x^2}{2}-\int \frac{1}{x}\cdot \frac{x^2}{2}dx$

$\Rightarrow I=\frac{x^2\ln x}{2}-\frac{1}{2}\int xdx\Rightarrow I=\frac{x^2\ln x}{2}-\frac{x^2}{4}+C$