Question Number 62596 by aliesam last updated on 23/Jun/19
$$\int\mathrm{sin}^{\mathrm{100}} \left(\mathrm{x}\right)\:\mathrm{cos}^{\mathrm{100}} \left(\mathrm{x}\right)\:\mathrm{dx} \\ $$
Answered by MJS last updated on 23/Jun/19
![∫sin^(100) x cos^(100) x dx=∫(sin x cos x)^(100) dx= =(1/2^(100) )∫sin^(100) 2x dx= [t=2x → dx=(dt/2)] =(1/2^(101) )∫sin^(100) t dt now use ∫sin^n t dt=−((cos t sin^(n−1) t)/n)+((n−1)/n)∫sin^(n−2) t dt](https://www.tinkutara.com/question/Q62607.png)
$$\int\mathrm{sin}^{\mathrm{100}} \:{x}\:\mathrm{cos}^{\mathrm{100}} \:{x}\:{dx}=\int\left(\mathrm{sin}\:{x}\:\mathrm{cos}\:{x}\right)^{\mathrm{100}} {dx}= \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}^{\mathrm{100}} }\int\mathrm{sin}^{\mathrm{100}} \:\mathrm{2}{x}\:{dx}= \\ $$$$\:\:\:\:\:\left[{t}=\mathrm{2}{x}\:\rightarrow\:{dx}=\frac{{dt}}{\mathrm{2}}\right] \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}^{\mathrm{101}} }\int\mathrm{sin}^{\mathrm{100}} \:{t}\:{dt} \\ $$$$\mathrm{now}\:\mathrm{use}\:\int\mathrm{sin}^{{n}} \:{t}\:{dt}=−\frac{\mathrm{cos}\:{t}\:\mathrm{sin}^{{n}−\mathrm{1}} \:{t}}{{n}}+\frac{{n}−\mathrm{1}}{{n}}\int\mathrm{sin}^{{n}−\mathrm{2}} \:{t}\:{dt} \\ $$