Q111 / 160 · AP EAMCET 2023 (17 May Shift 2) · Mathematics

$\int \frac{1-\cos x}{\cos x(1+\cos x)} d x=$

  • Correct answer
    A
    $\log |\sec x+\tan x|-2(\operatorname{cosec} x-\cot x)+C$
  • B
    $\log |\sec x+\tan x|-2(\operatorname{cosec} x+\cot x)+C$
  • C
    $\log |\sec x+\tan x|+2(\operatorname{cosec} x-\cot x)+C$
  • D
    $\log |\sec x+\tan x|+2(\operatorname{cosec} x+\cot x)+C$

Solution

$\int \frac{1-\cos x}{\cos x(1+\cos x)} d x=\int \frac{1+\cos x-2 \cos x}{\cos x(1+\cos x)} d x$ $\begin{aligned} & =\int \frac{1}{\cos x} d x-2 \int \frac{1}{1+\cos x} d x \\ & =\int \sec x d x-2 \int \frac{1}{1+\cos x} d x \\ & =\int \sec x d x-2 \int \frac{1-\cos x}{1-\cos ^2 x} d x \\ & =\int \sec x d x-2 \int\left(\operatorname{cosec}^2 x-\cot x \cdot \operatorname{cosec} x\right) d x \\ & =\log |\sec x+\tan x|-2[\operatorname{cosec} x-\cot x]+\mathrm{c}\end{aligned}$

Hence, the correct answer is option: 'A'.

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