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Question Number 97759 by bobhans last updated on 09/Jun/20

$$\int \frac{\sin {}^5\left(x\right)dx}{\sqrt{\cos \left(x\right)}}?$$

Answered by bemath last updated on 09/Jun/20

$$\int \frac{{(1-\cos {}^{2}x)}^2\sin xdx}{\sqrt{\cos x}}=$$$$set\sqrt{\cos x}=z\Rightarrow -\sin xdx=2zdz$$$$\int \frac{{(1-z^4)}^2}{z}(-2zdz)=$$$$-2\int (1-2z^4+z^8)dz=$$$$-2z+\frac{4}{5}{z}^5-\frac{2}{9}{z}^9+c=$$$$-2\sqrt{\cos x}+\frac{4}{5}\cos {}^{2}x\sqrt{\cos x}-\frac{2}{9}\cos {}^{4}x\sqrt{\cos x}+c$$

Commented by bobhans last updated on 09/Jun/20

$$\mathrm{alright}$$

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