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

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

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

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

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