Integration by parts question (1 Viewer)

[cutemouse]

Hello, could someone please help me with the following question by the integration by parts method?

Integrate[x(secx)(tanx)dx]

Thanks

 

[Trebla]

u = x, du = dx
dv = sec x.tan x dx, v = sec x
∫ x.sec x.tan x dx = x.sec x - ∫ sec x dx
= x.sec x - ln (sec x - tan x) + c

 

[cutemouse]

Hey Trebla, Thanks for that. Just wondering how come the integrated value of (secx.tan x) is (secx), and the integrated value of (sec x) is ln(sec x - tan x)

dv = sec x.tan x dx, v = sec x

Thanks again!

 

[tacogym27101990]

Hey Trebla, Thanks for that. Just wondering how come the integrated value of (secx.tan x) is (secx), and the integrated value of (sec x) is ln(sec x - tan x)



Thanks again!

d/dx(sec x) = d/dx((cosx)^-1)
= -(cosx)^-2 * -sinx
=sinx/cos^2x
=1/cosx* sinx/cosx
=secxtanx

therefore the integral of secxtanx is secx

 

[jpmeijer]

Isn't the integral of sec x = ln(sec x + tan x)?

You could differentiate ln(sec x + tan x) to check that its derivative is indeed sec x, just like tacoqym did for the other one.

Another way to look at the integral of sec x is to multiply top and bottom by (sec x + tan x), giving: [sec x(sec x + tan x)]/(sec x + tan x).
This is an integral in the form f'(x)/f(x). That is, the integral equals the logarithm of the bottom, ln(sec x + tan x).

Hope that made sense...

 

[cutemouse]

Hmm, the secxtanx is on the standard integrals table, should've looked first.

But I still don't know how I could get the Integrated value of secx or any other trig function involving logs.

Could someone tell me if there's a 'list' of these results that we should know?

Thanks

 

[tacogym27101990]

Hmm, the secxtanx is on the standard integrals table, should've looked first.

But I still don't know how I could get the Integrated value of secx or any other trig function involving logs.

Could someone tell me if there's a 'list' of these results that we should know?

Thanks

the integration of secx and whatnot are basically things you just have to know