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Question

Mathematics Question on integral

By using the properties of definite integrals, evaluate the integral: ∫01x(1−x)n dx\int^1_0x(1-x)^n\,dx

Answer

Let I=∫01x(1−x)n dx\int_{0}^{1} x(1-x)^n \,dx

∴I=∫01(1−x)(1−(1−x))n dx\int_{0}^{1} (1-x)(1-(1-x))^n \,dx

=∫01x(1−x)(x)n dx\int_{0}^{1} x(1-x) (x)^n \,dx

=∫01(xn−xn+1)dx=∫^1_0(x^n-x^{n+1})dx

=[xn+1n+1−xn+2n+2]     (∫0aƒ(x)dx=∫0aƒ(a−x)dx)=[\frac{x^{n+1}}{n+1}-\frac{x^{n+2}}{n+2}]\,\,\,\,\, (∫^a_0ƒ(x)dx=∫^a_0ƒ(a-x)dx)

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

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

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