Question

$\frac{dxe^{\left(2\cos(x)\right)}}{e^{\left(2\cos(x)\right)}+1}$​

Answer 1

♣ Qᴜᴇꜱᴛɪᴏɴ :

$\Large\boxed{\sf{\dfrac{d\:x\:e^{\left(2\:\cos \left(x\right)\right)}}{e^{\left(2\:\cos \left(x\right)\right)+1}}}}$

★═════════════════★  

♣ ᴀɴꜱᴡᴇʀ :

$\boxed{\sf{\dfrac{dx}{e}}}$

★═════════════════★

♣ ᴄᴀʟᴄᴜʟᴀᴛɪᴏɴꜱ :

$\sf{\dfrac{dxe^{2\cos \left(x\right)}}{e^{\left(2\cos \left(x\right)\right)+1}}}$

$\mathrm{Remove\:parentheses}:\quad \left(a\right)=a$

$\sf{=\dfrac{dxe^{2\cos \left(x\right)}}{e^{2\cos \left(x\right)+1}}}$

$\mathrm{Apply\:exponent\:rule}:\quad \dfrac{x^a}{x^b}=\dfrac{1}{x^{b-a}}$

$\sf{\dfrac{e^{2\cos \left(x\right)}}{e^{2\cos \left(x\right)+1}}=\dfrac{1}{e^{2\cos \left(x\right)+1-2\cos \left(x\right)}}}$

$\sf{\displaystyle=\frac{dx}{e^{2\cos \left(x\right)+1-2\cos \left(x\right)}}}$

$\mathrm{Add\:similar\:elements:}\:2\cos \left(x\right)+1-2\cos \left(x\right)=1$

$\boxed{\sf{=\dfrac{dx}{e}}}$