Question Number 21591 by Isse last updated on 29/Sep/17
$$\mathrm{5cos}\:^{\mathrm{5}} {t}\mathrm{sin}\:{t} \\ $$
Commented by mrW1 last updated on 29/Sep/17
$$\mathrm{what}'\mathrm{s}\:\mathrm{your}\:\mathrm{concrete}\:\mathrm{question}? \\ $$
Answered by $@ty@m last updated on 29/Sep/17
$${Let}\mathrm{cos}{t}={u}\: \\ $$$$\Rightarrow−\mathrm{sin}{tdt}={du}\: \\ $$$$\int\mathrm{5cos}\:^{\mathrm{5}} {t}\mathrm{sin}\:{tdt} \\ $$$$=\int\mathrm{5}{u}^{\mathrm{5}} \left(−{du}\right) \\ $$$$=−\mathrm{5}\int{u}^{\mathrm{5}} {du} \\ $$$$=−\mathrm{5}\frac{{u}^{\mathrm{6}} }{\mathrm{6}}+\mathrm{C} \\ $$$$=\frac{−\mathrm{5}}{\mathrm{6}}\mathrm{cos}^{\mathrm{6}} {t}\:+\mathrm{C} \\ $$