Question Number 208205 by efronzo1 last updated on 07/Jun/24
$$\:\:\:\int\:\left({x}^{\mathrm{3}} .\:\mathrm{5}^{\mathrm{2}{x}^{\mathrm{2}} −\mathrm{2}} \:\right)\:{dx}\:=? \\ $$
Answered by Frix last updated on 07/Jun/24
$$\int{x}^{\mathrm{3}} \mathrm{5}^{\mathrm{2}{x}^{\mathrm{2}} −\mathrm{2}} {dx}=\frac{\mathrm{1}}{\mathrm{25}}\int{x}^{\mathrm{3}} \mathrm{25}^{{x}^{\mathrm{2}} } {dx}\:\overset{{t}={x}^{\mathrm{2}} } {=} \\ $$$$\:\frac{\mathrm{1}}{\mathrm{50}}\int\mathrm{25}^{{t}} {tdt}\:\overset{\mathrm{by}\:\mathrm{parts}} {=}\:\frac{\mathrm{25}^{{t}} {t}}{\mathrm{100ln}\:\mathrm{5}}−\frac{\mathrm{1}}{\mathrm{100ln}\:\mathrm{5}}\int\mathrm{25}^{{t}} {dt}= \\ $$$$=\frac{\mathrm{25}^{{t}} {t}}{\mathrm{100ln}\:\mathrm{5}}−\frac{\mathrm{25}^{{t}} }{\mathrm{200}\left(\mathrm{ln}\:\mathrm{5}\right)^{\mathrm{2}} }= \\ $$$$=\frac{\mathrm{25}^{{x}^{\mathrm{2}} } \left(\mathrm{2ln}\:\mathrm{5}\:{x}^{\mathrm{2}} −\mathrm{1}\right)}{\mathrm{200}\left(\mathrm{ln}\:\mathrm{5}\right)^{\mathrm{2}} }+{C} \\ $$