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INTEGRALS — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsINTEGRALS3 Marks
Question
Evaluate the following definite integrals:
$\int_{1}^\limits{2}\text{e}^{2\text{x}}\Big(\frac{1}{\text{x}}-\frac{1}{2\text{x}^2}\big)\text{dx}$
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Answer

Let $\text{I}=\int_{1}^\limits{2}\text{e}^{2\text{x}}\Big(\frac{1}{\text{x}}-\frac{1}{2\text{x}^2}\big)\text{dx}$ Then,
$\text{I}=\int_{1}^\limits{2}\text{e}^{2\text{x}} \frac{1}{\text{x}}-\int_{1}^\limits{2}\text{e}^{2\text{x}} \frac{1}{2\text{x}^2}\text{ dx}$
Integrating first term by parts,
$\Rightarrow\text{I}=\bigg\{\Big[\frac{\text{e}^{2\text{x}}}{2\text{x}}\Big]^2_1-\int_{1}^\limits{2}-\text{e}^{2\text{x}} \frac{1}{2\text{x}^2}\bigg\}-\int_{1}^\limits{2}\text{e}^{2\text{x}} \frac{1}{2\text{x}^2}\text{ dx}$
$\Rightarrow\text{I}=\Big[\frac{\text{e}^{2\text{x}}}{2\text{x}}\Big]^2_1$
$\Rightarrow\text{I}=\frac{\text{e}^4}{4}-\frac{\text{e}^2}{2}$
$\Rightarrow\text{I}=\frac{\text{e}^4-2\text{e}^2}{4}$

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