Integration by Parts (1 Viewer)

[untouchablecuz]

Integrate:

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| (secx)^3 dx
/

Thanks in advance.

 

[Trebla]

Integrate:

/
| (secx)^3 dx
/

Thanks in advance.

u = sec x, du = sec x tan x dx
dv = sec²x dx, v = tan x

∫ 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 - sec x dx
2∫ sec³x dx = sec x tan x + ∫ sec x dx
∫ sec³x dx = (1/2) sec x tan x + (1/2) ln (sec x + tan x) + c

 

[3unitz]

u = sec x
u' = (sec x)(tan x)

v = tan x
v' = (sec x)^2

Int (sec x)^3 dx = (sec x)(tan x) - Int (sec x)(tan x)^2 dx

Int (sec x)^3 dx = (sec x)(tan x) - Int (sec x)[(sec x)^2 - 1] dx

Int (sec x)^3 dx = (sec x)(tan x) - Int (sec x)^3 dx + Int sec x dx

2 Int (sec x)^3 dx = (sec x)(tan x) + Int sec x dx

Int (sec x)^3 dx = [(sec x)(tan x) + ln |(sec x) + (tan x)|] / 2 + C