Question Number 84246 by pew_247 last updated on 10/Mar/20
$$\int\mathrm{cos}^{{n}} \:{mx}\:\:{dx}\:= \\ $$
Answered by TANMAY PANACEA last updated on 10/Mar/20
$${I}_{{n},{m}} =\int{cos}^{{n}} {mxdx}=\int{cos}^{{n}−\mathrm{1}} {mx}.{cosmxdx} \\ $$$${cos}^{{n}−\mathrm{1}} {mx}.\frac{{sinmx}}{{m}}+\int\left({n}−\mathrm{1}\right){cos}^{{n}−\mathrm{2}} {mx}.{msinmx}.\frac{{sinmx}}{{m}}{dx} \\ $$$$\frac{{cos}^{{n}−\mathrm{1}} {mxsinmx}}{{m}}+\left({n}−\mathrm{1}\right)\int{cos}^{{n}−\mathrm{2}} {mx}\left(\mathrm{1}−{cos}^{\mathrm{2}} {mx}\right) \\ $$$${I}_{{n},{m}} =\frac{{cos}^{{n}−\mathrm{1}} {mx}.{sinmx}}{{m}}+\left({n}−\mathrm{1}\right){I}_{{n}−\mathrm{2},{m}} −\left({n}−\mathrm{1}\right){I}_{{n},{m}} \: \\ $$$${I}_{{n},{m}} =\frac{{cos}^{{n}−\mathrm{1}} {mx}.{sinmx}}{{mn}}+\frac{{n}−\mathrm{1}}{{n}}{I}_{{n}−\mathrm{2},{m}} \\ $$$$\boldsymbol{{pls}}\:\boldsymbol{{check}} \\ $$$$ \\ $$