Question

`int e^x` sec x (1 + tan x) dx बराबर है:

पर्याय

• 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

स्पष्टीकरण:

माना `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`

यह मान समीकरण (1) में रखने पर,

`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`