Resolved: "integral of sec⁡xtan⁡x"

amevaa0y

Answered question

2022-01-23

integral of

\sec x \tan x

Micah May

Beginner2022-01-24Added 11 answers

integration by parts

\int \sec x \tan x d x = \int \sec^{2} x \sin x d x

Let

u = \sin x and d v = \sec^{2} x d x \Rightarrow d u = \cos x d x and v = \tan x

\int \sec^{2} x \sin x d x = \tan x \sin x - \int \tan x \cos x d x

= \tan x \sin x - \int \sin x d x

= \tan x \sin x + \cos x + c

note that

\tan x \sin x + \cos x = \frac{\sin^{2} x}{\cos x} + \frac{\cos^{2} x}{\cos x}

= \frac{1}{\cos x}

= \sec x

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