Evaluate The Integral Displaystyle Int12L (1/x - 1/2x2) e2Xdx Using Substitution.

Given: \(\int_{1}^{2} (\frac{1}{x} – \frac{1}{2x^{2}}) e^{2x} dx\)

Let 2x = t

= 2dx = dt

When x=1, t=2 and when x=2, t=4

\(\int_{1}^{2} (\frac{1}{x} – \frac{1}{2x^{2}}) e^{2x} dx = \int_{2}^{4} (\frac{2}{t} – \frac{2}{t^{2}}) e^{t} dt\\\Rightarrow \int_{4}^{2}(\frac{1}{t} – \frac{1}{t^{2}})e^{t}dt\\Let \frac{1}{t} = f(t)\\Then {f}'(t) = -\frac{1}{t^{2}}\\\Rightarrow \int_{2}^{4} (\frac{1}{t} – \frac{1}{t^{2}})e^{t}dt = \int_{2}^{4}e^{t}[f(t) + {f}'(t)]dt\\\Rightarrow [e^{t}f(t)]\frac{4}{2}\\\Rightarrow [e^{t} * \frac{1}{t}]_{2}^{4}\\\Rightarrow \frac{e^{4}}{4} – \frac{e^{2}}{2}\\\Rightarrow \frac{e^{2}(e^{2} – 2)}{4} \)

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