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INTEGRALS — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsINTEGRALS2 Marks
Question
Integrate the function in Exercise:

$\frac{2+\sin2\text{x}}{1+\cos2\text{x}}\text{e}^{\text{x}}$

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Answer

$\text{I}=\int\bigg(\frac{2+\sin2\text{x}}{1+\cos2\text{x}}\bigg)\text{e}^{\text{x}}$
$=\int\bigg(\frac{2+2\sin\text{x}\cos\text{x}}{2\cos^{2}\text{x}}\bigg)\text{e}^{\text{x}}$
$=\int\bigg(\frac{1+\sin\text{x}\cos\text{x}}{\cos^{2}\text{x}}\bigg)\text{e}^{\text{x}}$
$=\int\big(\sec^{2}\text{x}+\tan\text{x}\big)\text{e}^{\text{x}}$
$\text{Let}\ \text{f(x)}=\tan\text{x}\Rightarrow\text{f(x)}=\sec^{2}\text{x}$
$\therefore\text{I}=\int\big[\text{f(x)}+\text{f}'\text{(x)}\big]\text{e}^{\text{x}}\text{dx}$
$=\text{e}^{\text{x}}\ \text{f(x)}+\text{C}$
$=\text{e}^{\text{x}}\tan\text{x}+\text{C}$

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