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Question Number 21970 by j.masanja06@gmail.com last updated on 07/Oct/17

integrate  ∫sec^3 xdx

$${integrate} \\ $$$$\int{sec}^{\mathrm{3}} {xdx} \\ $$

Answered by Tikufly last updated on 07/Oct/17

 ∫sec^3 xdx=∫ (secx)(sec^2 x)dx  ∫sec^3 xdx=secxtanx−∫secxtan^2 xdx  ∫sec^3 xdx=secxtanx−∫secx(sec^2 x−1)dx  ∫sec^3 xdx=secxtanx−∫sec^3 xdx+∫secxdx  2∫sec^3 xdx=secxtanx+log∣secx+tanx∣    ∫sec^3 xdx=(1/2)secxtanx+(1/2)log∣secx+tanx∣+C

$$\:\int\mathrm{sec}^{\mathrm{3}} {xdx}=\int\:\left(\mathrm{sec}{x}\right)\left(\mathrm{sec}^{\mathrm{2}} {x}\right){dx} \\ $$$$\int\mathrm{sec}^{\mathrm{3}} {xdx}=\mathrm{sec}{x}\mathrm{tan}{x}−\int\mathrm{sec}{x}\mathrm{tan}^{\mathrm{2}} {xdx} \\ $$$$\int\mathrm{sec}^{\mathrm{3}} {xdx}=\mathrm{sec}{x}\mathrm{tan}{x}−\int\mathrm{sec}{x}\left(\mathrm{sec}^{\mathrm{2}} {x}−\mathrm{1}\right){dx} \\ $$$$\int\mathrm{sec}^{\mathrm{3}} {xdx}=\mathrm{sec}{x}\mathrm{tan}{x}−\int\mathrm{sec}^{\mathrm{3}} {xdx}+\int\mathrm{sec}{xdx} \\ $$$$\mathrm{2}\int\mathrm{sec}^{\mathrm{3}} {xdx}=\mathrm{sec}{x}\mathrm{tan}{x}+\mathrm{log}\mid\mathrm{sec}{x}+\mathrm{tan}{x}\mid \\ $$$$\:\:\int{sec}^{\mathrm{3}} {xdx}=\frac{\mathrm{1}}{\mathrm{2}}{secxtanx}+\frac{\mathrm{1}}{\mathrm{2}}{log}\mid{secx}+{tanx}\mid+{C} \\ $$

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