Question

`int e^x sec x(1 + tanx) dx` equals

Options

• `e^x cos x + c`

• `e^x sec x + c`

• `e^x sin x + c`

• `e^x tan x + c`

Answer 1

`e^x 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`  ......(i)

`I_1 = int e^x (sec x) e^x dx`

Taking sec `x` as I function and `e^x` as II function, integration by parts.

`I_1 = (sec x) int e^x - int (sec x tan x int e^x dx)`

= `(sec x)e^x - int e^x sec x tan x dx`

Putting this value in (i)

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`