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Indefinite Integration — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsIndefinite Integration2 Marks
Question
Evaluate the following : $\int\left(\frac{\cos x}{1-\cos x}\right) \cdot d x$
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Answer

$
\begin{aligned}
I & =\int\left(\frac{\cos x}{1-\cos x}\right)\left(\frac{1+\cos x}{1+\cos x}\right) \cdot d x \\
& =\int \frac{\cos x(1+\cos x)}{1-\cos ^2 x} \cdot d x \\
& =\int\left(\frac{\left.\cos x+\cos ^2 x\right)}{\sin ^2 x}\right) \cdot d x \\
& =\int\left(\frac{\cos x}{\sin ^2 x}+\frac{\cos ^2 x}{\sin ^2 x}\right) \cdot d x \\
& =\int\left(\operatorname{cosec} x \cdot \cot x+\cot ^2 x\right) \cdot d x \\
& =\int\left(\operatorname{cosec} x \cdot \cot x+\operatorname{cosec}^2 x-1\right) \cdot d x \\
& =(-\operatorname{cosec} x)+(-\cot x)-x+c \\
& =-\operatorname{cosec} x-\cot x-x+c
\end{aligned}
$

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