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Question Number 152420 by tabata last updated on 28/Aug/21

$${\int }_0^{\frac{\pi }{2}}\frac{e^{sinx}}{e^{cosx}+e^{sinx}}dx?$$

Answered by mindispower last updated on 28/Aug/21

$$f(a+b-x)+f\left(x\right)=c$$$${\int }_a^{b}f\left(x\right)dx=\frac{1}{2}c(b-a)$$

Answered by Lordose last updated on 28/Aug/21

$$\mathrm{\Omega }={\int }_0^{\frac{\mathit{\pi }}{2}}\frac{{\mathrm{e}}^{\sin \left(\mathrm{x}\right)}}{{\mathrm{e}}^{\cos \left(\mathrm{x}\right)}+{\mathrm{e}}^{\sin \left(\mathrm{x}\right)}}\mathrm{dx}\left(1\right)$$$$\mathrm{\Omega }\stackrel{{\mathbf{king}}^{\prime }\mathbf{s}\mathbf{rule}}{=}{\int }_0^{\frac{\mathit{\pi }}{2}}\frac{{\mathrm{e}}^{\cos \left(\mathrm{x}\right)}}{{\mathrm{e}}^{\cos \left(\mathrm{x}\right)}+{\mathrm{e}}^{\sin \left(\mathrm{x}\right)}}\mathrm{dx}\left(2\right)$$$$\frac{\left(1\right)+\left(2\right)}{2}::$$$$\mathrm{\Omega }=\frac{1}{2}{\int }_0^{\frac{\mathit{\pi }}{2}}1\mathrm{dx}=\frac{\mathit{\pi }}{4}$$

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