Let I=\(\int_{0}^{1} x(1-x)^n \,dx\)
∴I=\(\int_{0}^{1} (1-x)(1-(1-x))^n \,dx\)
=\(\int_{0}^{1} x(1-x) (x)^n \,dx\)
\(=∫^1_0(x^n-x^{n+1})dx\)
\(=[\frac{x^{n+1}}{n+1}-\frac{x^{n+2}}{n+2}]\,\,\,\,\, (∫^a_0ƒ(x)dx=∫^a_0ƒ(a-x)dx)\)
=\([\frac{1}{n+1}-\frac{1}{n+2}]\)
=\(\frac{(n+2)-(n+1)}{(n+1)(n+2)}\)
=\(\frac{1}{(n+1)(n+2)}\)
Evaluate:
$\displaystyle \int_{0}^{3} x \cos(\pi x) \, dx$
Find:$\displaystyle \int \dfrac{dx}{\sin x + \sin 2x}$