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Question Number 60311 by aliesam last updated on 19/May/19

$$\int \frac{dx}{\sqrt{sec{h}^2\left(x\right)+1}}dx$$

Commented by MJS last updated on 20/May/19

$$\mathrm{sech}x=\frac{1}{\cosh x}$$

Commented by maxmathsup by imad last updated on 20/May/19

$$whatissech\left(x\right)?...idontusethisnotation...$$

Answered by MJS last updated on 20/May/19

$$\int \frac{dx}{\sqrt{1+{\mathrm{sech}}^{2}x}}=\int \frac{\cosh x}{\sqrt{1+{\cosh }^{2}x}}dx=$$$$[t=\sinh x\to dx=\frac{dt}{\cosh x}]$$$$=\int \frac{dt}{\sqrt{t^2+2}}=\ln (t+\sqrt{t^2+2})=$$$$=\ln (\sinh x+\sqrt{2+{\sinh }^{2}x})+C$$

Commented by aliesam last updated on 20/May/19

$$\int \frac{dx}{\sqrt{1+{sech}^2\left(x\right)}}=\int \frac{cosh\left(x\right)}{\sqrt{1+{cosh}^2\left(x\right)}}$$

Commented by MJS last updated on 20/May/19

$$\mathrm{yes}\mathrm{but}{\mathrm{it}}^{\prime }\mathrm{s}\mathrm{just}\mathrm{a}\mathrm{typo}$$

Commented by aliesam last updated on 20/May/19

$$yes.thankyouzir$$

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