\(\frac{\frac{{x}^2 \cdot 3 + \left(2 + x \cdot 4\right)}{2}}{{\left(e^{x}\right)}^{\left(1 - \left(-1\right)\right)}}\)
- Started with
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
18.8
- Using strategy
rm 18.8
- Applied exp-neg to get
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot \color{red}{e^{-\left(1 + \varepsilon\right) \cdot x}}}{2} \leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot \color{blue}{\frac{1}{e^{\left(1 + \varepsilon\right) \cdot x}}}}{2}\]
18.8
- Applied un-div-inv to get
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{red}{\left(\frac{1}{\varepsilon} - 1\right) \cdot \frac{1}{e^{\left(1 + \varepsilon\right) \cdot x}}}}{2} \leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{\frac{\frac{1}{\varepsilon} - 1}{e^{\left(1 + \varepsilon\right) \cdot x}}}}{2}\]
18.8
- Applied exp-neg to get
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \color{red}{e^{-\left(1 - \varepsilon\right) \cdot x}} - \frac{\frac{1}{\varepsilon} - 1}{e^{\left(1 + \varepsilon\right) \cdot x}}}{2} \leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \color{blue}{\frac{1}{e^{\left(1 - \varepsilon\right) \cdot x}}} - \frac{\frac{1}{\varepsilon} - 1}{e^{\left(1 + \varepsilon\right) \cdot x}}}{2}\]
18.7
- Applied un-div-inv to get
\[\frac{\color{red}{\left(1 + \frac{1}{\varepsilon}\right) \cdot \frac{1}{e^{\left(1 - \varepsilon\right) \cdot x}}} - \frac{\frac{1}{\varepsilon} - 1}{e^{\left(1 + \varepsilon\right) \cdot x}}}{2} \leadsto \frac{\color{blue}{\frac{1 + \frac{1}{\varepsilon}}{e^{\left(1 - \varepsilon\right) \cdot x}}} - \frac{\frac{1}{\varepsilon} - 1}{e^{\left(1 + \varepsilon\right) \cdot x}}}{2}\]
18.8
- Applied frac-sub to get
\[\frac{\color{red}{\frac{1 + \frac{1}{\varepsilon}}{e^{\left(1 - \varepsilon\right) \cdot x}} - \frac{\frac{1}{\varepsilon} - 1}{e^{\left(1 + \varepsilon\right) \cdot x}}}}{2} \leadsto \frac{\color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 + \varepsilon\right) \cdot x} - e^{\left(1 - \varepsilon\right) \cdot x} \cdot \left(\frac{1}{\varepsilon} - 1\right)}{e^{\left(1 - \varepsilon\right) \cdot x} \cdot e^{\left(1 + \varepsilon\right) \cdot x}}}}{2}\]
23.7
- Applied simplify to get
\[\frac{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 + \varepsilon\right) \cdot x} - e^{\left(1 - \varepsilon\right) \cdot x} \cdot \left(\frac{1}{\varepsilon} - 1\right)}{\color{red}{e^{\left(1 - \varepsilon\right) \cdot x} \cdot e^{\left(1 + \varepsilon\right) \cdot x}}}}{2} \leadsto \frac{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 + \varepsilon\right) \cdot x} - e^{\left(1 - \varepsilon\right) \cdot x} \cdot \left(\frac{1}{\varepsilon} - 1\right)}{\color{blue}{{\left(e^{x}\right)}^{\left(1 - \left(-1\right)\right)}}}}{2}\]
23.7
- Applied taylor to get
\[\frac{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 + \varepsilon\right) \cdot x} - e^{\left(1 - \varepsilon\right) \cdot x} \cdot \left(\frac{1}{\varepsilon} - 1\right)}{{\left(e^{x}\right)}^{\left(1 - \left(-1\right)\right)}}}{2} \leadsto \frac{\frac{3 \cdot {x}^2 + \left(2 + 4 \cdot x\right)}{{\left(e^{x}\right)}^{\left(1 - \left(-1\right)\right)}}}{2}\]
5.4
- Taylor expanded around 0 to get
\[\frac{\frac{\color{red}{3 \cdot {x}^2 + \left(2 + 4 \cdot x\right)}}{{\left(e^{x}\right)}^{\left(1 - \left(-1\right)\right)}}}{2} \leadsto \frac{\frac{\color{blue}{3 \cdot {x}^2 + \left(2 + 4 \cdot x\right)}}{{\left(e^{x}\right)}^{\left(1 - \left(-1\right)\right)}}}{2}\]
5.4
- Applied simplify to get
\[\frac{\frac{3 \cdot {x}^2 + \left(2 + 4 \cdot x\right)}{{\left(e^{x}\right)}^{\left(1 - \left(-1\right)\right)}}}{2} \leadsto \frac{\left(x \cdot x\right) \cdot 3 + \left(2 + 4 \cdot x\right)}{2 \cdot {\left(e^{x}\right)}^{\left(1 - \left(-1\right)\right)}}\]
5.4
- Applied final simplification
- Applied simplify to get
\[\color{red}{\frac{\left(x \cdot x\right) \cdot 3 + \left(2 + 4 \cdot x\right)}{2 \cdot {\left(e^{x}\right)}^{\left(1 - \left(-1\right)\right)}}} \leadsto \color{blue}{\frac{\frac{{x}^2 \cdot 3 + \left(2 + x \cdot 4\right)}{2}}{{\left(e^{x}\right)}^{\left(1 - \left(-1\right)\right)}}}\]
5.4
