Q235 / 270 · NDA 2021 (Phase 1) · Mathematics

What is $\int \frac{d x}{\sec x+\tan x}$ equal to ?

  • A
    $\ln (\sec x)+\ln |\sec x+\tan x|+c$
  • B
    $\ln (\sec x)-\ln |\sec x+\tan x|+c$
  • C
    $\sec x \tan x-\ln |\sec x-\tan x|+c$
  • Correct answer
    D
    $\ln |\sec x+\tan x|-\ln |\sec x|+c$

Solution

$\begin{aligned} I & =\int \frac{d x}{\sec x+\tan x}=\int \frac{(\sec x-\tan x) \cdot d x}{\left(\sec ^2 x-\tan ^2 x\right)} \\ & =\int \sec x \cdot d x-\int \tan x \cdot d x \\ & =l n|(\sec x+\tan x)|-(-\operatorname{nn}|\cos x|)+C \\ & =l n|\sec x+\tan x|-l \operatorname{n}|\cos x|^{-1}+C \\ & =l n|\sec x+\tan x|-l \operatorname{n}|\sec x|+C\end{aligned}$

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

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