Evaluate : (\integral\_{0}^{1}x) (1 – x)99 dx | MATH - FUNDAMENTAL DEFINITE INTEGRATION, DEFINITE INTEGRATION BY SUBSTITUTION | SolveeIt

Evaluate : $\int_{0}^{1}x$ (1 – x)⁹⁹ dx

$\frac{1}{10100}$

$\frac{1}{11000}$

$\frac{1}{10010}$

None

Answer

$\frac{1}{10100}$

Explanation

Let I = $\int_{0}^{1}x$ (1 – x)⁹⁹ dx

= $\int_{0}^{1}{(1 - x)}$ (1 – (1 – x))⁹⁹ dx (By Prop. IV)

= $\int_{0}^{1}{(1 - x)}$ x⁹⁹ dx

= $\int_{0}^{1}{(x^{99} - x^{100})}$ dx

= $\left. \ \frac{x^{100}}{100} - \frac{x^{101}}{101} \right|_{0}^{1}$

= $\frac{1}{100}$ – $\frac{1}{101}$

= $\frac{1}{10100}$ .

Question: Evaluate : \(\int_{0}^{1}x\) (1 – x)<sup>99</sup> dx...