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INTEGRALS — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsINTEGRALS3 Marks
Question
Find the integrals of the functions in Exercises:
$\frac{\cos\text{x}}{1+\cos\text{x}}$
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Answer

$\frac{\cos\text{x}}{1+\cos\text{x}}=\frac{\cos^2\frac{\text{x}}{2}-\sin^2\frac{\text{x}}{2}}{2\cos^2\frac{\text{x}}{2}}$ $\ \ \ \ \ \ \ \ \bigg[\cos\text{x}=\cos^2\frac{\text{x}}{2}-\sin^2\frac{\text{x}}{2}\text{ and }\cos\text{x}=2\cos^2\frac{\text{x}}{2}-1\bigg]$
$=\frac{1}{2}\bigg[1-\tan^2\frac{\text{x}}{2}\bigg]$
$\therefore\int\frac{\cos\text{x}}{1+\cos\text{x}}\text{ dx}=\frac{1}{2}\int\bigg(1-\tan^2\frac{\text{x}}{2}\bigg)\text{ dx}$
$=\frac{1}{2}\int\bigg(1-\sec^2\frac{\text{x}}{2}+1\bigg)\text{dx}$
$=\frac{1}{2}\int\bigg(2-\sec^2\frac{\text{x}}{2}\bigg)\text{dx}$
$=\frac{1}{2}\Bigg[2\text{x}-\frac{\tan\frac{\text{x}}{2}}{\frac{1}{2}}\Bigg]+\text{C}$
$=\text{x}-\tan\frac{\text{x}}{2}+\text{C}$

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