Best solutions to - "Evaluate the integral: sec⁡x(sec⁡x+tan⁡x)dx"

Yolanda Jorge

Answered question

2021-11-20

Evaluate the integral:

\sec x (\sec x + \tan x) d x

soniarus7x

Beginner2021-11-21Added 17 answers

Step 1
To find the below integral.

\int \sec x (\sec x + \tan x) d x

Step 2
Using u substitution method

u = \sec x + \tan x

\Rightarrow d u = (\sec^{2} x + \tan x \sec x) d x

\Rightarrow d u = \sec x (\sec x + \tan x) d x

\Rightarrow d u = u \sec x d x

\Rightarrow \sec x d x = \frac{d u}{u}

\int \sec x (\sec x + \tan x) d x = \int u \frac{d u}{u}

\int \sec x (\sec x + \tan x) d x = \int d u

\int \sec x (\sec x + \tan x) d x = u + C

\int \sec x (\sec x + \tan x) d x = \sec x + \tan x + C

George Morin

Beginner2021-11-22Added 13 answers

Step 1: Expand.

\int \sec^{2} x + \sec x \tan x d x

Step 2: Use Sum Rule:

\int f (x) + g (x) d x = \int f (x) d x + \int g (x) d x

\int \sec^{2} x d x + \int \sec x \tan x d x

Step 3: The derivative of

\tan x i s \sec^{2} x

\tan x + \int \sec x \tan x d x

Step 4: The derivative of

\sec x i s \sec x \tan x

\tan x + \sec x

Step 5: Add constant.

\tan x + \sec x + C

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