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

(1) ∫ sin ((√x)) dx (2) ∫ cos ((√x) ) dx (3) ∫ tan ((√x) ) dx

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

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

∫sin((√x))dx x=t^2 ⇒1=2t(dt/dx) 2∫tsin(t)dt −2tcos(t)+2∫cost −2tcos(t)+2sint+C =−2(√x)cos((√x))+2sin((√x))+C

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

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

∫cos((√x))dx 2∫tcos(t) 2tsint−2∫sint 2tsint+2cost+C 2(√x)sin((√x))+2cos((√x))+C

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

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