Question

Evaluate: $\int x\frac{d}{dx}\left(x^2\ln x\right)dx$

• A. $\frac{x^2}{9}(\ln x+2)$
• B. $\frac{x^3}{9}(6\ln x+1)$
• C. $\frac{x^2}{9}(3\ln x+2)$
• D. $\frac{x^3}{3}(\ln x+2)$

Answer 1

Answer: (B)

$\frac{x^3}{9}(6\ln x+1)$

$u\frac{dv}{dx}=\frac{duv}{dx}-\frac{vdu}{dx}u=xv=x^2\ln xx\frac{d}{dx}\left(x^2\ln x\right)=\frac{d}{dx}(x\cdot x^2\ln x)-x^2\ln x\frac{dx}{dx}I=\int x\frac{d}{dx}\left(x^2\ln x\right)dx=\int [\frac{d}{dx}\left(x^3\ln x\right)-x^2\ln x\frac{dx}{dx}]dxI=\int \frac{d}{dx}\left(x^3\ln x\right)dx-\int x^2\ln xdx$

Using integration by parts;

$I=x^3\ln x-[\frac{x^3}{3}\ln x-\int \frac{x^3}{3}\frac{dx}{x}]+kI=\frac{2x^3}{3}\ln x+\left[\frac{x^3}{9}\right]+CI=\frac{x^3}{9}(6\ln x+1)+C$