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Question Number 115661 by bemath last updated on 27/Sep/20

$$\left(1\right)\int \sin \left(\sqrt{x}\right)dx$$$$\left(2\right)\int \cos \left(\sqrt{x}\right)dx$$$$\left(3\right)\int \tan \left(\sqrt{x}\right)dx$$

Answered by Dwaipayan Shikari last updated on 27/Sep/20

$$\int \sin \left(\sqrt{\mathrm{x}}\right)\mathrm{dx}\mathrm{x}={\mathrm{t}}^2\Rightarrow 1=2\mathrm{t}\frac{\mathrm{dt}}{\mathrm{dx}}$$$$2\int \mathrm{tsin}\left(\mathrm{t}\right)\mathrm{dt}$$$$-2\mathrm{tcos}\left(\mathrm{t}\right)+2\int \mathrm{cost}$$$$-2\mathrm{tcos}\left(\mathrm{t}\right)+2\mathrm{sint}+\mathrm{C}$$$$=-2\sqrt{\mathrm{x}}\cos \left(\sqrt{\mathrm{x}}\right)+2\sin \left(\sqrt{\mathrm{x}}\right)+\mathrm{C}$$

Answered by Dwaipayan Shikari last updated on 27/Sep/20

$$\int \cos \left(\sqrt{\mathrm{x}}\right)\mathrm{dx}$$$$2\int \mathrm{tcos}\left(\mathrm{t}\right)$$$$2\mathrm{tsint}-2\int \mathrm{sint}$$$$2\mathrm{tsint}+2\mathrm{cost}+\mathrm{C}$$$$2\sqrt{\mathrm{x}}\sin \left(\sqrt{\mathrm{x}}\right)+2\cos \left(\sqrt{\mathrm{x}}\right)+\mathrm{C}$$

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