The integral $$\int_{0}^{1} x(1 - x)^n \, dx \text{ is equal to:}$$​​

Correct option is B

We have 

$$\begin{aligned}I &= \int_{0}^{1} x(1 - x)^n dx \\I &= \int_{0}^{1} (1 - x)[1 - (1 - x)]^n dx \\I &= \int_{0}^{1} (1 - x) x^n dx = \int_{0}^{1} (x^n - x^{n+1}) dx \\I &= \left[ \frac{x^{n+1}}{n+1} - \frac{x^{n+2}}{n+2} \right]_0^1 \\I &= \left( \frac{1}{n+1} - \frac{1}{n+2} \right) - 0 = \frac{1}{(n+1)(n+2)}\end{aligned}$$​​