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Question
`int e^x sec x (1 + tan x) dx` equals:
Options
ex cos x + C
ex sec x + C
ex sin x + C
ex tan x + C
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Solution
ex sec x + C
Explanation:
Let `I = int e^x sec x (1 + tan x) dx`
`= int e^x (sec x + sec x tan x) dx`
` = int (sec x) e^x dx + int e^x sec x tan x dx`
`= I_1 + int e^x sec x tan x` .... (1)
`I_1 = int (sec x)e^x dx`
`I_1 = (sec x) int e^x dx - int (sec x tan x int e^x dx) dx`
`= (sec x) e^x - int e^x sec x tan x dx`
Putting this value in equation (1),
`I = I_1 + int e^x sec x tan x dx`
`= (sec x) e^x - int e^x sec x tan x dx + int e^x sec x tan x dx + C`
`= e^x sec x + C`
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