Question

By using the properties of definite integrals, evaluate the integral: \(\int^1_0x(1-x)^n\,dx\)

Answer 1

Let I=\(\int_{0}^{1} x(1-x)^n \,dx\)

∴I=\(\int_{0}^{1} (1-x)(1-(1-x))^n \,dx\)

=\(\int_{0}^{1} x(1-x) (x)^n \,dx\)

\(=∫^1_0(x^n-x^{n+1})dx\)

\(=[\frac{x^{n+1}}{n+1}-\frac{x^{n+2}}{n+2}]\,\,\,\,\, (∫^a_0ƒ(x)dx=∫^a_0ƒ(a-x)dx)\)

=\([\frac{1}{n+1}-\frac{1}{n+2}]\)

=\(\frac{(n+2)-(n+1)}{(n+1)(n+2)}\)

=\(\frac{1}{(n+1)(n+2)}\)