integrate[sec^3(x)]dx?

Question

show that
int(sec cubed x)dx
=(1/2){secx*tanx+
log(secx+tanx)}+C
using the integration by
parts method

good luck

Answer 1

let u = sec x dv = sec^2 x
then du = sec x tan x v = tan x

so int(sec^3 x dx) = sec x tan x - int(sec x tan^2 x dx)
= sec x tan x - int(sec x (sec^2 x - 1) dx)
= sec x tan x - int(sec^3 x dx) + int(sec x dx)
= sec x tan x - int(sec^3 x dx) + ln (sec x + tan x) + C

so now we have
int(sec^3 x dx) = sec x tan x - int(sec^3 x dx) + ln (sec x + tan x) + C

solve for int(sec^3 x dx):
2 int(sec^3 x dx) = sec x tan x + ln (sec x + tan x) + C
int(sec^3 x dx) = 1/2 (sec x tan x + ln (sec x + tan x)) + C

note that the end result C is different from the above C, but since it's an arbitrary constant it doesn't matter

Answer 2

Right. This is a bit tricky. To do it, you need to have quite a few facts ready.

1 sec^2 x = 1 + tan^2 x so tan^2 x = sec^2 x - 1

(this is one of the Pythagorean identities and is derived from cos^2 + sin^2 = 1)

2 The derivative of tanx is sec^2 x

so the integral of sec^2 x is tanx

3 The derivative of secx is secxtanx

4 The derivative of ln(secx + tanx) is secx

This is using the chain rule - but whenever you differentiate ln(f(x) you get f'(x)/f(x) so in this case you get:

(secx tanx+sec^2 x)/(secx+tanx) = secx(tanx+secx)/(secx+tanx) = secx

This gives you that the integral of secx is ln(secx + tanx)

5 Finally, you need to be familiar with the rule for integration by parts. I will use S for the integral sign. We are trying to find Suvdx, Assume that we can integrate v and differentiate u.

Suvdx = uSvdx - S(u'Svdx)dx + c

Right. Ready to start. (phew)

Let I = Ssec^3 x dx = S secx sec^2 x dx and we are going to integrate the sec^2 x and differentiate the secx. So

I = tanx secx - S(secx tanx tanx)dx + c
= tanx secx - S(secx tan^2 x) dx + c
= tanx secx - S(secx (sec^2 x - 1))dx + c
= tanx secx - Ssec^3 x dx + Ssecx dx + c
= tanx secx - I + ln(secx + tanx) + c

Take the I to the other side and divide both sides by 2 to give:

I = 0.5(tanx secx + ln(secx + tanx)) + c as required

Hope this makes sense!