Question Number 82617 by necxxx last updated on 23/Feb/20
$$\int\mathrm{sin}\:\left(\mathrm{101}{x}\right)\left({sinx}\right)^{\mathrm{99}} {dx} \\ $$
Answered by TANMAY PANACEA last updated on 23/Feb/20
$$\int{sin}\left({n}+\mathrm{1}\right){xsin}^{{n}−\mathrm{1}} {xdx}\:\:\:\:{n}=\mathrm{100} \\ $$$$\int\left({sinnxcosx}+{cosnxsinx}\right){sin}^{{n}−\mathrm{1}} {xdx} \\ $$$$\int\left({sinnxcosxsin}^{{n}−\mathrm{1}} {x}+{cosnxsin}^{{n}} {x}\right){dx} \\ $$$$\int\left({sinnx}×\frac{{d}\left({sin}^{{n}} {x}\right)}{{dx}}×\frac{\mathrm{1}}{{n}}+{sin}^{{n}} {x}×\frac{{d}\left({sinnx}\right)}{{dx}}×\frac{\mathrm{1}}{{n}}\right){dx} \\ $$$$\frac{\mathrm{1}}{{n}}\int\frac{{d}}{{dx}}\left({sinnx}.{sin}^{{n}} {x}\right){dx} \\ $$$$\frac{\mathrm{1}}{{n}}\left({sinnx}.{sin}^{{n}} {x}\right)+{c} \\ $$$$\frac{\mathrm{1}}{\mathrm{100}}\left({sin}\mathrm{100}{xsin}^{\mathrm{100}} {x}\right)+{c} \\ $$$$ \\ $$
Commented by peter frank last updated on 23/Feb/20
$${help}\:{qn}\:\mathrm{74411} \\ $$
Commented by TANMAY PANACEA last updated on 23/Feb/20
$${ok}\:{let}\:{me}\:{see} \\ $$