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Question Number 143952 by akolade last updated on 19/Jun/21

Answered by Olaf_Thorendsen last updated on 20/Jun/21

$$\mathrm{C}=\int \frac{\cosh x}{\cosh x+\sinh x}dx$$$$\mathrm{S}=\int \frac{\sinh x}{\cosh x+\sinh x}dx$$$$\mathrm{C}+\mathrm{S}=\int dx=x+\mathrm{cst}\left(1\right)$$$$\mathrm{C}-\mathrm{S}=\int \frac{\cosh x-\sinh x}{\cosh x+\sinh x}dx$$$$\mathrm{C}-\mathrm{S}=\int \frac{{(\cosh x-\sinh x)}^2}{{\cosh }^{2}x-{\sinh }^{2}x}dx$$$$\mathrm{C}-\mathrm{S}=\int {(\frac{e^x+e^{-x}}{2}-\frac{e^x-e^{-x}}{2})}^{2}dx$$$$\mathrm{C}-\mathrm{S}=\int e^{-2x}dx=-\frac{1}{2}{e}^{-2x}+\mathrm{cst}\left(2\right)$$$$\frac{\left(1\right)+\left(2\right)}{2}:\mathrm{C}=-\frac{1}{4}{e}^{-2x}+\frac{1}{2}x+\mathrm{cst}$$

Answered by Olaf_Thorendsen last updated on 20/Jun/21

$$\mathrm{C}=\int \frac{\cosh x}{\cosh x+\sinh x}dx$$$$\mathrm{C}=\int \frac{\frac{e^x+e^{-x}}{2}}{\frac{e^x+e^{-x}}{2}+\frac{e^x-e^{-x}}{2}}dx$$$$\mathrm{C}=\frac{1}{2}\int \frac{e^x+e^{-x}}{e^x}dx$$$$\mathrm{C}=\frac{1}{2}\int (1+e^{-2x})dx$$$$\mathrm{C}=\frac{1}{2}(x-\frac{1}{2}{e}^{-2x})+\mathrm{cst}$$

Commented by akolade last updated on 20/Jun/21

$$\mathrm{OR}$$$$\frac{{2\mathrm{xe}}^{2\mathrm{x}}-1}{{4\mathrm{e}}^{2\mathrm{x}}}$$

Answered by mathmax by abdo last updated on 20/Jun/21

$$\mathrm{\Phi }=\int \frac{\mathrm{ch}\left(\mathrm{x}\right)}{\mathrm{ch}\left(\mathrm{x}\right)+\mathrm{sh}\left(\mathrm{x}\right)}\mathrm{dx}\Rightarrow \mathrm{\Phi }=\int \frac{\frac{{\mathrm{e}}^{\mathrm{x}}+{\mathrm{e}}^{-\mathrm{x}}}{2}}{\frac{{\mathrm{e}}^{\mathrm{x}}+{\mathrm{e}}^{-\mathrm{x}}}{2}+\frac{{\mathrm{e}}^{\mathrm{x}}-{\mathrm{e}}^{-\mathrm{x}}}{2}}\mathrm{dx}$$$$=\int \frac{{\mathrm{e}}^{\mathrm{x}}+{\mathrm{e}}^{-\mathrm{x}}}{{2\mathrm{e}}^{\mathrm{x}}}\mathrm{dx}=\frac{1}{2}\int (1+{\mathrm{e}}^{-2\mathrm{x}})\mathrm{dx}=\frac{\mathrm{x}}{2}+\frac{1}{2}\int {\mathrm{e}}^{-2\mathrm{x}}\mathrm{dx}$$$$=\frac{\mathrm{x}}{2}-\frac{1}{4}{\mathrm{e}}^{-2\mathrm{x}}+\mathrm{K}$$

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