Question

$$\int x{(1-x)}^{23}dx$$

 

Answer 1

$$\int x{(1-x)}^{23}dx$$
$$\text{Let 1 - x }=t$$
$$\Rightarrow x=1-t$$
$$\Rightarrow 1=-\frac{dt}{dx}$$
$$\Rightarrow dx=-dt$$
$$Now,\int x{(1-x)}^{23}dx$$
$$=-\int (1-t)\cdot t^{23}dt$$
$$=-\int (t^{23}-t^{24})dt$$
$$=\int (t^{24}-t^{23})dt$$
$$=\frac{t^{25}}{25}-\frac{t^{24}}{24}+C$$
$$=\frac{24t^{25}-25t^{24}}{600}+C$$
$$=\frac{t^{24}}{600}[24t-25]+C$$
$$=\frac{{(1-x)}^{24}}{600}[24(1-x)-25]+C$$
$$=\frac{-1}{600}{(1-x)}^{24}(1+24x)+C$$