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Question Number 70719 by oyemi kemewari last updated on 07/Oct/19

∫sin (101x)sin^(99) x dx

$$\int\mathrm{sin}\:\left(\mathrm{101x}\right)\mathrm{sin}\:^{\mathrm{99}} \mathrm{x}\:\mathrm{dx} \\ $$

Answered by mind is power last updated on 07/Oct/19

sin(101x)=sin(100x)cos(x)+sin(x)cos(100x) ⇒∫sin(101x)sin^(99) (x)=∫sin(100x)cos(x)sin^(99) (x)+∫cos(100x)sin^(100) (x) ∫sin(100x)cos(x)sin^(99) (x)=((sin^(100) (x))/(100)).sin(100x)−∫cos(100x)sin^(100) (x) ⇒∫sin(100x)cos(x)sin^(99) (x)+∫cos(100x)sin^(100) (x)=((sin^(100) (x)sin(100x))/(100))+c

$${sin}\left(\mathrm{101}{x}\right)={sin}\left(\mathrm{100}{x}\right){cos}\left({x}\right)+{sin}\left({x}\right){cos}\left(\mathrm{100}{x}\right) \\ $$$$\Rightarrow\int{sin}\left(\mathrm{101}{x}\right){sin}^{\mathrm{99}} \left({x}\right)=\int{sin}\left(\mathrm{100}{x}\right){cos}\left({x}\right){sin}^{\mathrm{99}} \left({x}\right)+\int{cos}\left(\mathrm{100}{x}\right){sin}^{\mathrm{100}} \left({x}\right) \\ $$$$\int{sin}\left(\mathrm{100}{x}\right){cos}\left({x}\right){sin}^{\mathrm{99}} \left({x}\right)=\frac{{sin}^{\mathrm{100}} \left({x}\right)}{\mathrm{100}}.{sin}\left(\mathrm{100}{x}\right)−\int{cos}\left(\mathrm{100}{x}\right){sin}^{\mathrm{100}} \left({x}\right) \\ $$$$\Rightarrow\int{sin}\left(\mathrm{100}{x}\right){cos}\left({x}\right){sin}^{\mathrm{99}} \left({x}\right)+\int{cos}\left(\mathrm{100}{x}\right){sin}^{\mathrm{100}} \left({x}\right)=\frac{{sin}^{\mathrm{100}} \left({x}\right){sin}\left(\mathrm{100}{x}\right)}{\mathrm{100}}+{c} \\ $$

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