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Question Number 77323 by Boyka last updated on 05/Jan/20

Commented by Tony Lin last updated on 05/Jan/20

$$\int \frac{1}{\sqrt{cos(x-\frac{\pi }{2})}}dx$$$$=\int \frac{1}{\sqrt{1-2{sin}^2\left(\frac{2x-\pi }{4}\right)}}dx$$$$letu=\frac{2x-\pi }{4},\frac{du}{dx}=\frac{1}{2}$$$$2\int \frac{1}{\sqrt{1-2{sin}^{2}u}}du$$$$=2F(\frac{2x-\pi }{4}\mid 2)$$$$F(\theta \mid k)=\int \frac{1}{\sqrt{1-{ksin}^2\theta }}d\theta$$

Commented by MJS last updated on 05/Jan/20

$$\mathrm{this}\mathrm{leads}\mathrm{to}\mathrm{an}\mathrm{incomplete}\mathrm{elliptic}\mathrm{integral}$$$$\mathrm{of}\mathrm{first}\mathrm{order}$$$$\int \frac{dx}{\sqrt{\sin x}}=2\mathrm{F}(\frac{x}{2}-\frac{\pi }{4}\mid 2)$$$$\mathrm{look}\mathrm{it}\mathrm{up}\mathrm{on}\mathrm{the}\mathrm{web}$$

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