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Question Number 121466 by sdfg last updated on 08/Nov/20

$$\int \sqrt{cos\left(x\right)dx}$$

Commented by MJS_new last updated on 08/Nov/20

$$\int \sqrt{\cos xdx}\mathrm{or}\int \sqrt{\cos x}dx?$$

Commented by sdfg last updated on 08/Nov/20

$$\int \sqrt{\cos \mathrm{x}}\mathrm{dx}$$

Answered by MJS_new last updated on 08/Nov/20

$$\int \sqrt{\cos x}dx=$$$$[t=\frac{x}{2}\to dx=2dt]$$$$=2\int \sqrt{\cos 2t}dt=2\int \sqrt{{2\cos }^{2}t-1}dt=$$$$=2\int \sqrt{1-{2\sin }^{2}t}dt=2\mathrm{E}(t\mid 2)=$$$$=2\mathrm{E}(\frac{x}{2}\mid 2)+C$$

Commented by MJS_new last updated on 08/Nov/20

$$\int \sqrt{\sin x}dx=\int \sqrt{\cos (x-\frac{\pi }{2})}dx$$$$\Rightarrow$$$$\int \sqrt{\sin x}dx=2\mathrm{E}(\frac{x}{2}-\frac{\pi }{4}\mid 2)+C$$

Commented by MJS_new last updated on 08/Nov/20

$$\left(\mathrm{Elliptic}\mathrm{Integral}\right)$$

Commented by peter frank last updated on 08/Nov/20

$$\int \sqrt{\sin \mathrm{x}}dx=?$$

Commented by peter frank last updated on 08/Nov/20

$$\mathrm{thank}\mathrm{you}$$

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