Int tanx sec^2x dx can be taken as (by putting it in form of Int xdx)

Question

Int secx.secx tanx dx=(sec^2x)/2
Int tanx.sec^2x dx=(tan^2)/2.
Which one is correct and why is the difference?

Answer 1

I am sure you have seen the variation of the Pythagorean identity,

1 + tan^2 x = sec^2 x

so y = 1+tan^2 x and y = sec^2 x are one and the same function

for the first version:
y = 1 + tan^2 x
dy/dx = 2(tanx)(sec^2 x)

for the second version:
y = sec^2 x
dy/dx = 2(secx)(secx tanx) = 2 tanx sec^2 x
which is the same as the first result.

so naturally, from
y = 1 + tan^2 x
dy/dx = 2(tanx)(sec^2 x)
then ∫2(tanx)(sec^2 x) dx = tan^2 x + c

and from
y = sec^2 x
dy/dx = 2(secx)(secx tanx) = 2 tanx sec^2 x
then ∫ 2(sec^2 x tanx) dx = sec^2 x + k

where c and k are constants.

notice the two results are the same in their non-constant components

Answer 2

Both are correct. (The variation in the answers depends on what "u" to be substituted)

Integral (sec^2 (x) tan x) dx
This can be rewritten as:
Integral (sec(x)*sec(x)*tan(x)) dx
Here we use substitution. We let
u = sec(x)
du = sec(x)*tan(x) dx
Substituting,
Integral (u du)
= (1/2)(u^2) + C
= (1/2)(sec^2 (x)) + C

The other solution:
Integral (sec^2 (x) tan x) dx
Here, we let
u = tan(x)
du = sec^2 (x) dx
Substituting,
Integral (u du)
= (1/2)(u^2) + C
= (1/2)(tan^2 (x)) + C
But note the pythagorean identity: 1 + tan^2 (x) = sec^2 (x)
We substitute it here:
= (1/2)(sec^2 (x) - 1) + C
= (1/2)(sec^2 (x)) - 1/2 + C
= (1/2)(sec^2 (x)) + C

Hope this helps~ :3

Answer 3

Thank you very much. I had missed out the point on the constants of integration.