Evaluate the following integrals:

Let I $=\int \mathrm{x}(1-\mathrm{x})^{23} \mathrm{dx}$

Substituting $1-x=t \Rightarrow d x=-d t$

$\Rightarrow I=-\int(1-t) t^{23} d t$

$\Rightarrow I=-\int\left(t^{23}-t^{24}\right) d t$

$\Rightarrow I=-\left[\frac{t^{24}}{24}-\frac{t^{25}}{25}\right]+c$

$\Rightarrow I=\frac{(1-x)^{25}}{25}-\frac{(1-x)^{24}}{24}+c$

$\Rightarrow I=\frac{1}{600}(1-x)^{24}[24(1-x)-25]$

$\Rightarrow I=-\frac{1}{600}(1-x)^{24}[1+24 x]+c$