Evaluate ∫ (cosec^2 x + sec^2 x) dx.

Question 6

MediumOriginal question

Evaluate $\int \left(\operatorname{cosec}^{2} x + \sec^{2} x\right)\,dx$ .

\tan x + \cot x + C

Correct answerYour answer

\sec x\tan x + \operatorname{cosec} x\cot x + C

Correct answerYour answer

\tan x - \cot x + C

Correct answerYour answer

\cot x - \tan x + C

Correct answerYour answer

Show the answer and explanation

CorrectNot quiteAnswer C

Correct answer: C

$\frac{d}{dx}\tan x = \sec^2 x$ and $\frac{d}{dx}\cot x = -\operatorname{cosec}^2 x$ .
So $\int \sec^2 x\,dx = \tan x$ and $\int \operatorname{cosec}^2 x\,dx = -\cot x$ : the sum is $\tan x - \cot x + C$ .

Worth noting

$\tan x + \cot x$ misses the minus sign: the derivative of $\cot x$ is negative.

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