Question

Integration of sec^3xdx

Answer 1

Hello users .....

Solution:-
We know that:
∫ sec²x.dx = tanx + c
And
d(sec x) / dx = sec x . tan x
And
∫ Sec x .dx = ln | sec x + tan x | + c
And
∫ u.v.dx = u. ∫ v .dx -∫ {du/ dx .∫ v.dx }

Here,.
∫ sec³x .dx =∫ sec x . sec²x . dx

=>∫ sec³x .dx = sec x . ∫ sec²x .dx - ∫ {d(sec x)/dx . ∫ sec²x .dx}

=>∫ sec³x .dx = sec x . tan x - ∫(sec x . tanx). tan x.dx

=>∫ sec³x .dx = sec x . tan x - ∫sec x . tan²x .dx

=>∫ sec³x .dx = sec x .tan x - ∫ sec x ( sec²x - 1) .dx

=>∫ sec³x .dx = sec x .tan x - ∫ sec³x .dx + ∫ sec x .dx

=> 2∫ sec³x .dx = sec x .tan x + ln | sec x + tan x | + C

=>∫ sec³x.dx = 1/2 [ sec x .tan x + ln | sec x + tan x | ] + C Answer

Where, C is arbitrary constant of integration .

⭐✡ Hope it helps ✡⭐